ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR AUTOMORPHISMS OF THE PROJECTIVE UNITARY GROUPS By JOHN H. WALTER Project 2200 DETROIT ORDNANCE DISTRICT, ORDNANCE CORPS, U.S. ARMY CONTRACT DA-20-018-ORD-13281, DA Project No. 599-01-004 ORD Project No. TB2-001-(1040), OOR Project No. 31-124 April, 1954

PREFACE I would like to acknowledge my indebtedness to Professor Jean Dieudonne, at whose suggestion I investigated this problem. His encouragement and criticism have proved invaluable and have made the writing of this dissertation a valuable educational experience, To Professor Leonard Tornheim, I owe my gratitude for the help he has given me in improving the clarity and organization of this work and the care he has exercised in checking the results, This dissertation was written under the sponsership of the Office of Ordinance Research, U.S. Army, Contract DA-20-O18-ORD 13281. Professor Robert M. Thrall was the project director, ii

TABLE OF CONTENTS Page 1 INTRODUCTION CHAPTER I. PRELIMINARIES 1.1 Definitions of Certain Classical Groups 1.2 Classification of the Involutions of PUn(Kf) 1.3 Definition of the Invariants vu and mc 1,4 Characterizations of Subspaces 1.5 Involutions of the First Kind 1.6 Involutions of the Second Kind 1.7 Involutions of the Third Kind 1.8 Involutions of the Fourth Kind 1.9 A Theorem on Cyclic ummer Fields 1.10 Some Working Lemmas 3 3 6 7 8 10 10 11 17 19 19 CHAPTER II. INVOLUTIONS OF THE FIRST KIND 2.1 Extremal Involution Commuting with an Involution of the First Kind 2.2 Extremal Involution Commuting with an Involution of the Third Kind 2.3 Nonextremal Involutions of the First Kind 23 25 27 29 CHAPTER III. INVOLUTIONS OF THE THIRD KIND 3.1 Commuting Projective Involution whose is of the Second Kind 3.2 Commuting Projective Involution whose is of the Third Kind 3.3 Commuting Projective Involution whose is of the Fourth Kind 3.4 Anticommuting Projective Involutions. 3.5 Anticommuting Projective Involutions. Restriction Restriction Restriction Case I Case II 30 54 36 37 CHAPTER IV. INVOLUTIONS OF THE FOURTH KIND 4.1 Projective Involutions of the First Kind in Um(L,g) 4.2 Projective Involutions of the Second Kind in Um(L,g) 4.3 Projective Involutions of the Third Kind in Um(L,g) 4.4 Projective Involutions of the Fourth Kind in Um(L,g) 4.5 Projective Semilinear Involutions in rUm(L,g). Case I 4.6 Projective Semilinear Involution in FUm(L,g). Case II 40 40 42 44 47 49 51 iii

TABLE OF CONTENTS (Continued) Page CHAPTER V. CHARACTERIZATION OF EXTREMAL 5.1 Extremal Involutions 5.2 Involutions of the Second Kind 5.3 Involutions of the Third Kind 5.4 Involutions of the Fourth Kind 5.5 Principal Theorem INVOLUTIONS 53 53 55 58 60 62 APPENDIX. DETERMINATION OF THE AUTOMORPHISMS BIBLIOGRAPHY 63 iv

INTRODUCTION This dissertation is devoted to the determination of the automorphisms of the projective unitary groups. As such it is a contribution to the work on the general problem of determining the automorphisms of the classical groups. In the case of real or complex vector spaces, this problem has been solved for some time by infinitesimal methods. To 0. Schreier and B. van der Waerden [8] goes the credit for the first algebraic attack on this problem.1 Following them, J. Dieudonne greatly extended their results (e.g., see [5j) to many more classes of groups and to cases where, the vector space was in some cases over an arbitrary sfield (division ring or skew-field). As a general rule, the more complicated the group, the more restrictions Dieudonne placed on the sfield until, when dealing with the projective unitary groups, he required that the sfield be a finite field. Dieudonne's methods turn on obtaining a group-theoretical characterization of a certain type of involution and then applying the fundamental theorem of projective geometry. Dieudonne relies heavily on the structure of the groups to obtain his results. C. Rickart ([9] and [10]) broke away from this by using a device which had its origin in a paper by G. Mackey [7] and was able to obtain a more uniform method of attack on the problem. For the most part, Rickart dealt with the cases that Dieudonne had already considered. His method generalized to infinite dimensional spaces over sfields and did not need Dieudonne's requirement that the index of the form be greater than zero in the case of unitary and orthogonal groups.'Numbers in brackets refer to the bibliography at the end of this dissertation. 1

2 By refining the method of Rickart, we have been able to solve the problem for the projective unitary groups. We have also made use of Dieudonne's characterization of the centralizer of an involution. The problem is solved providing that the dimension of the underlying vector space is greater than 4 and not 6 and that the sfield is not of charcteristic 2 and not the finite field of three elements. For low dimensions, the methods of Dieudonne, Rickart, and myself all fail. In this connection, see Hua [6]. Along with Dieudonne and Rickart, we have approached the problem by studying the way the automorphisms transform the involutions of a projective unitary group. We have distinguished a certain class of involutions, which are called "extremal involutions". They are defined geometrically in terms of the way they act on the underlying vector space. Their importance comes from the fact that they are associated with the 1-dimensional subspaces of the underlying vector space. Our principal result is the characterization of these involutions in terms of the multiplicative structure of the projec — tive unitary group (Theorem 5.5)- Once this is done, the arguments of Rickart and Dieudonne apply. A summary of them is given in the appendix.

CHAPTER I PRELIMINARIES2 1.1 Definitions of Certain Classical Groups Let E be a right vector space of dimension n over a sfield K. The group of semilinear transformations3 of E onto itself is denoted by rLn(K) and is called the group of collineations. The set of transformations hX, such that h%(x) = xX with X in K for all x in E, forms the subgroup Hn(K) of homothetic mappings; hX is semilinear with respect to the inner automorphism X: 0 + = x 0 X. Sometimes the transformation hk will be denoted by x and hx(x) by X(x). Denote by P(E) the (n-l)-dimensional projective space formed by taking the 1-dimensional subspaces of E as points. The quotient group PfLn = rLn(K)/Hn can be identified as the group of projective collineations acting on P(E). It is the factor group of FLn which acts faithfully on P(E). 2This chapter takes its point of departure from Dieudonne's treatise, [2], Chapter I. Many statements are given here without proof. For these proofs and for more details concerning the various groups of transformations and the characterization of the various kinds of involutions which we shall introduce, the reader is referred to the aforementioned treatise. 3A function t of E into itself is semilinear with respect to an automorphism T of K if t(x) is additive in x and t(xo) = t(x)0T for 0 in K. 5

4 Suppose, furthermore, that K possesses an involutive4 anti-automorphism J. Let f be a nondegenerate hermitian sesquilinear form5 defined on E with respect to J. The subgroup rUn(K,f) of rLn(K) consisting of elements t such that f(t(x),t(y)) = etf(xy)T where T is the automorphism of K associated with t and et is an element of K, the "multiplier of t", depending only on t, is called the group of semisimilitudes. The fact that f is hermitian implies that eJ = et. The normal subgroup Un(K,f) of [-Un(Kf) consisting of linear transformations with multiplier 1 is the unitary group. For t in Un(Kf), we have the familiar relation f(t(x),t(y)) = f(x,y). The subgroups PrUn(Kf) and PUn(K,f) corresponding to rUn(K,f) and Un(K,f) under the natural homomorphism of:Ln(K) onto PrLn(K) are called, respectively, the group of projective semisimilitudes and the projective unitary group. They can be identified as the factor groups rUn(K,f)/Hn(K) and Un(K,f)/Hn(K) Un(K,f). With one exception, the group Hn(K) Un(K,f) is the center Cn of Un(K,f), as is shown, for example, in [4]. This exception, which occurs when K is the galois field GF(3) and n = 2, is excluded from the groups we shall consider. The center Cn consists of those homothetic transformation hi for which X is in the center Z of K and XJ% = 1o The corresponding set of such elements in Z forms 4In the case that K is commutative, J is naturally an automorphism, which may be the identity. 5A function f: E x E + K is said to be sesquilinear with respect to an anti-automorphism J, if f(x,y) is additive in x and y and f(xa,y3) XJf(x,y)P for c and P in K. A sesquilinear form f is said to be hermitian if f(y,x) = f(x,y)J and is said to be nondegenerate if f(x,y) = 0 for x in E and all y in E implies that x = 0.

5 a multiplicative subgroup, which we shall denote by C. The centralizer of Un(K,f) in rLn(K,f) is Hn(K). This will be shown in Section 1.10. With certain restrictions on the dimension n of E and on the sfield K, we shall determine the automorphisms of PUn(Kpf) by characterizing certain involutions of FUn(K,f) called extremal involutions, which are associated with 1-dimensional subspaces of E. These are defined in Section 1.5. Once this is done, known methods of Dieudonn4 [5] and Rickart 110] will yield our result (cf. Appendix). As Rickart points out, we can obtain even more information. Indeed, we can conclude that if PUm(L,g) is another projective unitary group acting on a right L-space F of dimension m, then n = m, K and L are isomorphic, and f and g are related by a given isomorphism between E and F (cf. Rickart [10], Theorems 4.1 and 4.3 or Theorem A.2 of this dissertation). Throughout this dissertation we restrict ourselves to the case where K is not of characteristic 2 and is not the finite field GF(3). We assume that the dimension n of E is finite as a matter of convenience. Our result extends easily and directly to the infinite dimensional case. It is to be noted that we include in our considerations the projective orthogonal groups POn(Kf), which are the specializations of the projective unitary groups to the case where K is commutative and J is the identity. A sesquilinear form f is said to be reflexive if f(y,x) = 0 implies that f(x,y) = O. With the exception of the case where f is a skew-symmetric bilinear form, which can occur only when K is commutative, Birkhoff and von Neumann [1] have shown that there exists an element d in K such that the sesquilinear form g = df is hermitian. It then follows that Un(K,f) = Un(K,g). So our result extends to this case also.

6 In passing, we also note that if f is hermitian and dJ = d, then g = df is again hermitian. But if dJ = -d, then g = df is skew-hermitian; that is, g(y,x) = -g(xy)J. 1.2 Classification of the Involutions of PUn(K,f) We shall use the methods of linear algebra and for that reason we look upon an element t in PUn(Kf) as a coset in Un(K,f) and deal with the elements t of T. In particular, if f is an involution of PUn(Kf), then an element u of Un(Kf) in U is called a projective involution. If u is an involution in PrUn(K,f), then u will be called a semilinear involution and u of rUn(K,f) in the coset u will be called a projective semilinear involution. The relation u2= 7 with 7 in Cn characterizes projective involutions. We shall now distinguish various types of projective involutions u where u2 = 7 in Cn: (i) u is of the first kind if = X where X is in Z and.X = 1. (ii) u is of the second kind if 7 = X2 where X is in Z and XJX = -1. (iii) u is of the third kind if y = k2 where X is not in Z. (iv) u is of the fourth kind if y is not a square in K. Note that if 7 = \2 and X is in Z, then /7 = 1 implies that (XJ)Z = 1 and thus that XJX = + 1. So in this case u is of either the first or second kind. If the projective involution u is of the kth kind, then so is every projective involution Tu with T in Cn. So we can say that the involution T of PUn(K,f) is of the kth kind. An extremal involution is a particular type of involution of the first kind * Because the identity 1 can obviously be distinguished from other involutions of the first kind, we shall exclude it from the set of involutions of the first kind and the elements of Cn from the set of projective involutions of the first kind.

7 1,3 Definition of the Invariants v- and ac u u We shall characterize the extremal involutions of PUn(K,f) by using group theoretical invariants similar to those introduced by Mickey [7]. Let J be a set of ihvolutionm.of PUn(K,f) and define c(2) to be those involutions of PUn(K,f) which commute with every element of. In particular if, U and V are distinct commuting involutions in PUn(K,f), denoted'.by v(u,v) the number of elements in c(c(UV)) v. the maximum of v(IV) for all involutions V f U u such that -V = Vi, # 1, and V # 1. In the case of the unitary and general linear groups, Rickart ([9] and [10]) characterized extremal involutions by what corresponds to our v-. It is easily seen that c(c(ui,V)) always contains the elements T, i, v, and uv. So the minimum of v- is at least 4. Rickart showed that an involution is u extremal if and only if vu = 4 in the cases he considered. In the case of the projective unitary groups, this is no longer true. So we carry Rickart's method one step further* Let d denote the, set of involutions U of PUn(Kf) for which v- = 4. We shall show that & contains the extremal involutions along u with the involutions of the third and fourth kinds. If 1, 7, and W are distinct mutually commuting involutions in, denote by a(u,v,w) the number of elements in c(c(U,Vw)) a- the maximum of t(iu,V) for all V and w subject to the condition that, V, and W are distinct mutually commuting involutions in J. Now the minimal value of n- is at least 8. We shall show that o- = 8 if U U and only if U is extremal with two possible exceptions;.viz., when n = 8 or 12, cases which we shall handle by other means, It is important to

8 note that it is necessary to require that v and w be in,Jin order to obtain our result. For this reason, we will devote the first part of this dissertation to the characterization of J. If U and V are commuting involutions of PUn(Kf) and u is in -. - 2 2 u and v is in v such that u = 7 in Cn and v = B in Cn, we have uv avu with a in Cno Multiplying on the left and right by u and v, respectively, we obtain Py = auvuv = a27. So a2 = 1 and a = + 1, This means that u and v commute or anticommute. Replacing u by Qu or v by qv with 1 in Cn does not change the value of a. So the same result holds for any involution u in U and any involution v in V. It is convenient to denote the set of elements of Un(Kf) determined by a set J in PUn(Kf) by A. Conversely, if J is a set of elements in Un(K,f), we write J for the set of elements in PUn(Kf) which has representatives in1. Likewise, when is a set of projective involutions c(J) and c(c()), denote those projective involutions determined respectively by c(T) and c(c(J)). Then c(J) can be characterized as the set of projective involutions which commute or anticommute with the elements of i. 1.4 Characterizations of Subspaces In the discussion of the different types of projective involutions we use some fundamental concepts and properties of subspaces connected with a hermitian sesquilinear form f. A vector x is said to be isotropic if f(x,x) = 0 and is said to be nonisotropic otherwise. A vector y is orthogonal to a vector x if f(xy) = 0. If S is a subspace, the orthogonal subspace S. is the subspace consisting of all the vectors orthogonal to every vector of S. A nonisotropic subspace S is a subspace

9 such that S^S1 contains only the zero vector, and otherwise S is said to be an isotropic subspace. If S C SL, then S is a totally isotropic subspace. In particular, the space E is nonisotropic since f is nondegenerate. If every vector of a basis for S is nonisotropic, then the basis is said to be nonisotropic. A nonisotropic subspace possesses an orthogonal nonisotropic basis and any subset of orthogonal nonisotropic vectors can be extended to such a basis. We denote the dimension of a subspace S by dim S or, if there should be any confusion, by dimK S. Then dim S + dim SL = n. The index v of the form f is the maximum of the dimensions of the totally isotropic subspaces of E. Every totally isotropic subspace is contained in a maximal totally isotropic subspace of dimension v. If [r] denotes the greatest integer less than or equal to a rational number r, v < [n/2]. In particular, when n is even and v = m = n/2, E = S T (direct sum) where S and T are totally isotropic subspaces. A 2-dimensional subspace or plane, as it is sometimes called, generated by two nonisotropic nonorthogonal isotropic vectors is nonisotropic, for its orthogonal subspace is complementary. Now let S and T be two maximal totally isotropic subspaces such that E = SG T. If x is in S, then SC (xK)I. So T! (xK)L and there exists a vector y in T such that f(x,y) t 0. Thus the plane generated by x and y will be nonisotropic, as its orthogonal subspace is complementary. By replacing y by y0 with 0 in K, we can take f(x,y) to be any nonzero element in Ko

10 1:5 Involutions of the First Kind Let U be a projective involution of the first kind and choose a representative u in u. Then u2 - 7 and 72 = 2 where X is in C. The projective involution k-lu is also in U and is easily seen to be an involution; so we can assume that u is an involution; that is, = 1. Then E is the direct sum E = U+ @ U- of two subspaces such that u(x) = + x according as x is in U-, respectively. Since, for x in U and y in U-, f(x,y) = f(u(x),u(y)) = -f(x,y) = 0, we have U+ and U orthogonal. Because they are complementary, they are nonisotropic. If dim U = p, u is said to be a p-involution. As u and -u are both in U, we can always take p < [n/2]. Then i will also be called a p-involution. Since we do not consider 1 as an involution of the first kind, p > 0. If p = 1 and u a projective involution of the first kind, then we say that u is a minimal involution and that U is an extremal involution. Our aim is to characterize the extremal involutions group-theoretically. It is readily verified that an element t of Un(K,f) or even of riL(K) commutes with u if and only if t(U+) = U+ and t(U-) = U-T Similarly t anticommutes with u if and only if t(U+) = U- and t(U-) = U+. In this latter case we have dim U+ = dim U- = n/2 = m. That is no element t of FLn(Kf) can anticommute with u unless u is an m-involution. 1.6 Involutions of the Second Kind Let u be an involution of the second kind in PUn(K,f) and choose a representative u in i. Then u2 = 7 where 7 = k2 with X in Z and AJi = -1. Again form u' = X-lu. Then u' is an involution in rUn(Kf) with multiplier -1 and E = U+ ( U where U+ and U- are the positive and negative eigenspaces of u'. Also u(x) = + xX according

11 as x is in U, respectively. Let x and y be both in U+ or both in U. Then f(xjy) = f(u'(x),u(y)) = -f(x,y) = O; so U+ and U- are totally isotropic. As E = U U, ddim U = m = n/2 and this case can occur only when n is even and the index v of f is m. The conditions for an element t of rLn(K) to commute or anticommute with u are the same as for projective involutions of the first kind. Let now u and v be two commuting projective involutions of the first or second kind. Then v leaves U+ and U- invariant. So if x is in U+ or U-, x + v(x) is in U+ or U-, respectively. It is easily seen that x + v(x) is in V+ or V-, respectively. As x= x+v(x) + x-v(x),we _ 2 2 have U+ = U+nV+ (B U+ nV and U- = U-V+ U- V-; so write E = P1 ~ PP 0 P3S P4 where P1 = Un V+, P2 = U+V-, P3 = U- nV, and P4 = U-nV-. Now suppose that u and v are of the first kind. Then if x is in P1 and y is in P2, P3, or P4, then u(y) = -y or v(y) = -y. Hence, as f(x,y) = f(u(x),u(y)) = f(v(x),v(y)), f(x,y) = -f(x,y) = O. This means that P1 is othogonal to P2 ~ PS3 P4. Hence P1 is nonisotropic. Similarly P2, P3, and P4 are nonisotropic and all the Pi's are orthogonal. 1.7 Involutions of the Third Kind Let u be an involution of the third kind and choose u in u. Then u2 = 7 in Cn where 7 = X2 and X is not in Z. We treat the following generalization, which arises in subsequent developments. Let a be an involutive automorphism of K and let u be a semilinear projective involution of rUn(Kf) relative to the automorphism a with multiplier e, such that u2 = in Cn with 7 = XX. In case ~ = X, we furthermore assume that x is not in Z.

12 Then u' = k-1u is a semilinear involution in rUn(K,f) with respect to the automorphism at = %-ro =a and with multiplier e' = (XJ)-le. It is readily verified that a'2 = 1 and a' ~ 1. Let K1 be the subsfield of K consisting of elements left fixed by a'. Then if 0 is in K, 0 = 0 +' + where 0 + a is in K1 and 0 - is in the 2 2 left and right Kl-subspace K1' consisting of elements Br of K such that Bri =-). If Q is an arbitrary element of K1', then K1' = K1Q = QK1 and K = K1 e KlQ. Thus E can be considered as a right vector space E* over K1 and dimK E* = 2n, As a transformation of E*, u' is a linear involution. So E* = U+ @ U where U+ and U_ are the positive and negative eigenspaces of u', respectively. If x is in IU, xQ is in U- and vice versa. So U = U+Q and E* = U+ 9 U+Q; that is, a Kj-basis for U+ is a K-basis for E. An element t of rUn(K,f) relative to an automorphism T where X = X and with multiplier et, which commutes with u, commutes also with hx-l and thus with u'. Considering the effect of u'tandtu' on a vector x0 with 0 in K, we can conclude that O'T = Ta' and so K1T = K1. Thus t can be restricted to a semilinear transformation t1 acting on U+. Hence, we can reduce the analysis of such transformations t to that of transformations acting on U. But in order to do this effectively, we have to consider t1 as an element of some group rUn(Kl,fi). Unfortunately the form f cannot in general be restricted to a sesquilinear form acting on U+. The principal hindrance is that K1 is not necessarily left invariant by Jo Indeed, it can be shown that 0Ja = e-lJe for 0 in K1 (Dieudonne [4], Section 14). Dieudonne shows that e can always be taken to be 1 by multiplying the form f by a scalar d'l such that e = d1a and dJ = + d. In fact, if e / -1, it suffices to take d = 1 + e. If e = -1 and there exists an element 4 of K1' such that

13 j J -*t take d = $+Jr. In both of these cases dJ = d, and so the form g = d-1f will be a hermitian sesquilinear form relative to the antiautomorphism T = Jd: 0 + -0 = d-l 0Jd However, if J = -r for all elements * of K1' and if 0 is any element of K1, we have (0*)J = -0 = rJJ = -0J. So J = r-1t. This means that the restriction of J is an automorphism of K1; hence K1 is commutative. Now let d be any element of K1'. Then da = -d; so e = -1 = d-'. But now g = d-if is a skew-hermitian sesquilinear form relative to the anti-automorphism T = Jd, However, 0T = d-10Jd = d-20d2 for 0 in K1 as d is in K1'. But d2 is in K1, which is commutative; so 0T = 0. Thus in this case the restriction of the anti-automorphism T to Ki is the identity. In the second paragraph below, we shall show that this situation does not occur. In any case, g is a nondegenerate hermitian or skew-hermitian sesquilinear form defined on E and u' is a semilinear involution in rUn(K,g) with multiplier 1. It is easily seen that Un(K,g) = Un(K,f) and rUn(Kf) = rUn(Kg). Furthermore now K T = K1; and, if x and y are in U+, g(x,y) = g(u'(x),u'(y)) = g(xy)a' and so is in K1. Thus the restriction fl of g to U+4 is a hermitian or skew-hermitian sesquilinear form such that f 1 U+ x U+ - K1. As a Kl-basis for tU is a K-basis for E, f is nondegenerate. So we can now form the group Un(Ki,f1) acting on the Kl-space U+t In the case that fl is skew hermitian, we have seen that the restriction of T is the identity and K1 is commutative. This means that fl is a skew symmetric bilinear form and Un(Kifl) is actually the symplectic group Spn(Kl,fl)o However, then every vector x of ITis such that f1(x,x) = 0. But f1(x,x) = d-lf(x,x) and since a K1-basis for U+

14 is a K-basis for E, U+ must contain a vector x such that f(x,x) # 0. Hence we have a contradiction; so we can always suppose that there exists an element r in K1' such that J i -4 and thus fl will always be a hermitian sesquilinear form. Now let us return to the consideration of the elements t which commute with u and which were previously described. Since g is a scalar multiple of f, rUn(K,f) = rUn(K,g) and Un(K,f) = Un(K,g). Therefore t can be considered to be in rUn(K,g) and, if t is linear, to be in Un(K,g). As an element of rUn(K,g), t has multiplier et' = d- etdo Now g(u'(t(x),u'(t(y))) et g(xy) = g(t(u'(x)), t(u'(y))) = et g(x,y)a'T. Since Ta' = a'T, it follows that et' = et' and hence et' is in K1. Then clearly the restriction tl of t to U+' is an element of rUn(Kl,fl). If t is in Un(K,f), tl is in Un(Klfl). The element tl can/be extended to the original transformation t acting on E be setting t(xO) = t1(x)QT for x in U+t This defines t on U-, which is what is needed, as E = U+QU-. Not every element of [Un(Klfl) can be extended to a transformation of rUn(K,f) in general.6 However, if tl is in Un(Kl,fl), the above process will always lead to a transformation t in Un(K,f) of which t, is the restriction. If tl is a projective involution of the kth kind in Un(Ki,fi), t will be a projective involution of the kth kind in Un(K,f). To summarize, we have the following result. Let u be a projective semilinear involution in rUn(K,f) relative to an automorphism a of K such that u2 = 7 in Cn where 7 = LXa with X in K. 6Cf. Dieudonne' [2], Section 14.

15 Assume furthermore that X is not in Z if X =. Set a' = a'1 and let K, be the fixed subsfield of K b] a' and E1 be the vector space E considered as a right Kl-space Then E = U ~ U where u(x) = + xX for x in IU, respectively. An element t of rUn(K,f) relative to an automorphism T of K such that kT = X can be restricted to a transformation t, in VUn(Klfl). If t is in Un(K,f), tl is in Un(Ki,fl). An element t1 of Un(Kifl) can be extended to an element of t in Un(K,f). If t, is a projective involution of the first kind in Un(Ki,fi), t is of the first kind in Un(K,f), and conversely. In certain cases we must discuss projective semilinear involutions which anticommute with u. In order to do this, we must first further investigate the structure of the sfield K,. Let Z, be the center of K, and let 0 be the automorphism of K given as 0 0 + + = Q-'1. Since (Q2) = 02 92 is in K1. Furthermore, for 0 in K1, 0a = 0; so leaves K, and Z1 invariant. Hence, the restriction of 0 to Z,, which we again denote by 0, is an involutive automorphism of Z,. Thus either every element of Z, commutes with 9 (in the case that the restriction of 0 to Z, is the identity) or, by galois theory, Z, = Zo ~ Zo,' where every element of Zo commutes with 9 and X1k = -1'. Let Z' be the subfield of Z, which is left fixed by 0 (this is Z, in the first case and Zo in the second). Then every element of Z' commutes with every element of K, and with Q. Since K = K1 D K1~, this means that Z' C Z; hence Z' C ZK1,o On the other hand KnZ C Z'. Therefore, Z' = ZnK1 and we have the following result. If 0 is the identity on Z1, Z, = KrZn.Z If is an involution on Z, with fixed subfield Zo, then ZO = KlnZa

In the important case where u is a projective involution of the third kind in Un(K,f), at = kA' and so X is in Z1. As K1 is the fixed subsfield of K by a' = -, Z C K1. So 0 is an involution on Zo = K nZ = Z and we can take I' = X. Therefore Z1 = Z G ZX; that is, Z1 = Z(X). In general, when a' leaves Z1 fixed and 0 does not, we have Z1 = Z ~ ZX'. In this case K1 is an interior galois subsfield of K and, according to a result of Dieudonne ([5], p.182), K = K1 ~ K1* where 4 is an element of K whose square =2 = a is in Z. However, this implies that *a = -4 Indeed, rt = a + b* with a and b in K1. Therefore, r = 2 = a(b+l) + b2r. Thus either a = 0 or b = -1. In the first case, b = + 1. But b i 1 as * is not in K1; so b = -1 and f' -f. In the second case a' a - But (at)2 = 2a = a' On the other hand,' (*a )2 = (a - $)2 = a2 + 12 - (*a + ar). Then a = a2 + a. So a = 0 and fr1 = -to Since Q was chosen arbitrarily in K1', we can henceforth assume Q = f, so that Q2 = a is in Z and 2 = 1. Now we consider the case that t in FUn(K,f) anticommutes with u where u is a projective involution of the third kind in FUn(K,f). Let T be the automorphism of K corresponding to t and et be the multiplier of t. Assume that XT = X, Then t commutes with h.-1 and thus anticommutes with u'. As 0a = -G, hQ anticommutes with u'o So hit is a projective semilinear involution of rUn(K,f) relative to the automorphism TQ which commutes with u', The multiplier of h t in rUn(K,g) is d-1 JetOd which again must be in K1 by the argument given previously in this section, Furthermore K1TG = K1; so hnt can be restricted to a transformation tl of rUn(Kl,fl) acting on U+. We say in this case that tl represents ta To summarize we have the following result.

17 Let u be a projective involution of the third kind in Un(Kf) and t a semilinear transformation of rUn(K,f) with respect to the automorphism T of K which anticommutes with u. Let u2 = y where y = X2 and assume furthermore that X = \) Then t is represented by a transformation tl in rUn(K, fl), which is semilinear relative to the automorphism TQ. 1.8 Involutions of the Fourth Kind Let U be an involution of the fourth kind in PUn(K,f) and choose u in U. Then u2 = 7 in Cn where now 7 is not a square in K. Again we ccnsider a generalization, which arises in subsequent developments. Let a be a given involutive automorphism of K and take u to be a projective semilinear involution in rUn(K,f) relative to the automorphism a and with multiplier e such that u2 = 7 where 7 is in C and is not of the form Xa for X in K. Dieudonne ([2], Section 13) has shown that there exists a quadratic extension L of K obtained by adjoining a square root p of 7 such that 0p = pOa for 0 in K. Obviously L is not commutative when a / 1, even though K may be. So the consideration of sfields is necessary at this point. Now E can be considered as a vector space F over L by setting xp u(x). The L-dimension of F is m = n/2. Consequently this type of projective involution exists only when n is even. The anti-automorphism J can be extended to L by setting pJ = peay7 = es-O Corresponding to f, we define a new nondegenerate hermitian sesquilinear form g by g(x,y) = f(x,y) + pf(xu(y))a 7- In FUm(L,g) u is a homothetic transformation hp with multiplier e.

18 It is easily seen that if t is an element of rUn(K,f) which is relative to an automorphism t of K and which commutes with u, t can be considered to be in rUm(L,g) relative to the extension of the automorphism T to L obtained by setting pT = p. Conversely, if t is an element of rUm(L,g) which is relative to an automorphism T of L leaving K invariant, t can be considered as an element of rUn(K,f) relative to the restriction of v to K which commutes with u. Let a be the automorphism of L defined by the relation a: p + -p. Clearly a commutes with any automorphism T which leaves invariant the fixed field K of a. Therefore, if t is an element of rUn(Kf) which is relative to an automorphism T of K and which anticommutes with u, t can be considered to be in [Um(L,g) relative to the automorphism aT of L. Conversely, any element t of [LUm(L,g) which is relative to an automorphism of the form aT where pT = p and KT = K can be considered as an element of FUn(Kf) which is relative to the automorphism T of K and which anticommutes with u. In the later chapters, it is necessary to identify the center ZL of L when u is a projective involution of the fourth kind in Un(Kf), that is, when e = 1 and a = 1. Then clearly p is in ZL; so Z(p) C ZL. On the other hand, if 0 X [ + pX is in ZL with E and n in K, g and T must commute with every element of K. This implies that E and r} are in Z; so ZLC Z(p). Consequently, the center ZL of L is the quadratic extension Z(p) of Z when u is a projective involution of the fourth kind in Un(K,f). In Section 4.2, we make reference to the case where Un(Kf) is the orthogonal group On(Kf) and u is a projective involution of the fourth kind in On(Kf). Then K is commutative and J is the identity by

19 definition of On(Kf). Also u2 = y is in Cn. But now J7Y = 72 = 1; so 7 = + 1. As 7 is not a square in K, y = -1, p = i, and L = K(i) where i = v-. In this case, the extension of the identity automorphism J of K to L is obtained by setting iJ = i-1 = -i. 1.9 A Theorem on Cyclic Kummer Fields The following well-known theorem on cyclic Kummer fields is useful in the discussion of projective involutions. Let Z be a field containing the primitive nth roots of unity and Z' the splitting field of an irreducible equation xn-c~ i O. Then if B is an element of Z' such that pn is in Z, either P is in Z or p = Xp where X is a root of xn-a and p is in Z. We shall apply the theorem when n = 2. in which case the proof is trivial. Since the characteristic of the fields with which we deal: is always different from 2. our hypothesis will be always satisfied. 1.10 Some Working Lemmas LEMMA 1.1 If K is not of characteristic 2, then for every pair of orthogonal nonisotropic vectors x and y in E and for any a in K, at least one of the vectors x + y(a-l), x + ya, and x + y(a+l) is nonisotropic. Compute first (1.1) f(x+yP,x+yP) = f(x,x) + Jf(yy). The lemma is trivial if a = 0 or a = + 1. So suppose that a i 0 and a # + 1 and that x+ya and x+y(a+l) are all isotropic. Then (1.1) vanishes for = a and = a + 1. Setting f(y,y) = J # 0, we have -f(x,x) = cisc = (a+1) (a ) +l)

20 From this it follows that = = +(a;J_ + a) Since | ~ 0, this is impossible and proves the lemma. LEMMA 1.2 Let r be a projective semilinear involution in run(Kf) relative to an automorphism a of K such that r2 = e with e in Z and a2 = 1. Let Ko be the subsfield of K left fixed. y a. Assume that Ko is not of characteristic 2 and is not the finite field GF(3). Let S be a nonisotropic subspace of dimension greater than 1. If for every nonisotropic vector x in S, r(x) = xb with 5x in K, then r(x) = x5 for every nonisotropic vector x in S where a is in K and is independent of x and a is the inner automorphism 8. 0+ 8 -0 +l. It is an immediate consesquence of thes lemma that if a = 1, 8 is in Z. Proof: Let xo be an arbitrary nonisotropic vector in S. Then r(xo) = xobxo~ Set 8 = 5x0 Let 51 be the inner automorphism 8s:~ 0 + 08 = 805 -land let K1 be the subsfield of K left fixed by E = $1a1' Now r2 = e implies 58a =, which is in Z. From this it follows that %2 = 1. By the argument of Section 1.7, [K:K1] = 1 or 2 according as XK = 1 or not, Now K1 is not the field GF(3); for, if K is infinite, this is impossible as [K2K1] < 2. If K is finite, it is commutative; so 5 = 1, t = a, and K = K0, which is not GF(3) by assumption. Let xo' be a nonisotropic vector of S orthogonal to xo. One always exists as S is nonisotropic of dimension greater than 1. Then r(Xo1) = Xo'0 o,, Again set =xo, = 5'o By Lemma 1.1 there exists an element a in K1 such that x, + xo'a is also nonisotropic. Because K1 GF(3), we can always choose a M 0 by choosing a in Lemma 1.1 not to be

21 0 or + 1. Then r(xo+Xo'a) = (xo+Xotc) xo,+ On the other hand, r(xO+XO') = r(xo) + r(xo1 )c. From these two equations we conclude that Xo(8xo+xo a-) = x('a -,o+x'a ) * Since x0 and xo' are linearly independent with respect to the sfield K, it follows that 8 = 8Xo+x0 t and then that'Gaa = a5. But ca = a as a is in K1. So ca = QG = b la8 and baa = a&. Consequently, b'cl = 50a and 8' = 5. However, xo' was chosen to be an arbitrary nonisotropic vector in S orthogonal to xo. Hence 5 = 8x o0 where 0 is in Ko Thus r(x0) = xo'08 = r(xol)Oa = Xo'l0a. So 08 = 80a and -a = 8-' = 08. Consequently, r(xo0) = r(xo)Oa = Xo0bO = x,08o As S in nonisotropic, there exists an orthogonal nonisotropic basis,x1X1,Y..,Xs-. for S where dim S = s. Then if x is any vector in S, x = Xoco+X1cZ+...+xs-Cs-.s r(x) = r(xoo) + r(xlal) +.. + r(xs-las-l) = x8, which completes the proof of the lemma. LEMMA 1.3 Let S be a subspace of E of dimension greater than 1 and let t be an element of rUn(K,f) relative to an automorphism a of K. Then if t(x) = xbx with 5x in K for every vector x of S, t(x) = x8 with 8 in K where 5 is independent of x in S and a is the inner automorphism:': 0 + 0 6 = -z0b. Let x be an arbitrary vector in So Then r(x) = xbx. For convenience, set & = 5x. Let y be a vector in S which is linearly independent of x. Then we have r(y) = yby and r(x+y) = (x+y) x+y. But r(x+y) = x8x + ybyo So 8x = bx+y = 8y. Similarly =x = y8x for 0 in K. So as argued in the proof of Lemma 1.2, 0a = -108 and r(x)= xS for all x in S where, in this case, it is only necessary to choose

22 a linearly independent basis for S and not an orthogonal nonisotropic bases. This proves the lemma. In particular, if an element r in r1Un(Kf) relative to an automorphism a of K commutes with every element of Un(K,f), it commutes with every 1-involution. Hence r leaves all 1-dimensional nonisotropic subspaces invariant. Then by Lemma 1,2, r(x) = x5 for all x in E where 8 is in K. Furthermore,a is the inner automorphism 8 of K; so r is in Hn. It is easily seen that every element of Hn is in the centralizer of Un(Kf) in rUn(Kf). Thus the centralizer of Un(K,f) in rUn(Kf) is Hn and the center of Un(Kf) is HnrUn(K^f) = Cn.

CHAPTER II INVOLUTIONS OF THE FIRST KIND In this chapter u will represent an involution of PUn(Kf) of the first kind. We shall assume first that n > 3. Later on we shall take n > 4 and exclude the case n = 6. The usual restrictions on the sfield K shall hold (cf Section 1.1). In accordance with Section 1.5, we choose the representative u of u which is a p-involution of Un(K,f) with p~ [n/2]1 Then E = U+ DUU where Us and U` are the positive and negative subspaces of Erespectively. We shall show that v= = 4 if u is a 1-involution, that it, extremal, while v- > 4 if u is a p-involution with p > 1. Let v be an involution of PUn(Kf) commuting with u. If u is extremal, we have LEMMA 2.1 If u is an extremal involution and v an involution commuting with u, then a projective involution v in v must commute with any projective involution u in u. Furthermore v is either of the first or third kind. As we remarked in Section 1.5> v cannot anticommute with u in u as dim U = 1 while dim E = n > 2. Consequently v(U+) = U+; so if x is in U+, v(x) = xyL with p, in K. But then v2(x) = xp. = xp; so P = p.2 and v is not of the fourth kind. If v is of the first or second kind, we have E = V+ GV and, as in Section 1.6, E = P1 &P2 s-P3 ~P4 and U+ = P1 0P2 = IU+V+ V V.V-. But, as U+ is 1-dimensional, U+ is contained + V+ V + in V or V-; that is, V or V contains a nonisotropic vector and consequently cannot be totally isotropic. Thus v is not of the second kind. 23

24 2.1 Extremal Involution Commuting with an Involution of the First Kind Let u be an extremal involution and u a 1-involution. Let v be an involution of the first kind distinct from u and commuting with u and v a p-involution in v with p [n/2]. Assume n > 3. We then have E = P1 6P2 < P3 ) P4 where the subspaces Pi are nonisotropic, or 0dimensional. Indeed, as dim U+ = 1, one of P1 = U +V+ or P2 = U Vmust contain only the 0 vector. Since u i 7, U- t V*, and U- = P @ P4 where dim P3 > 0 and dim P4 > 0. Thus E is decomposed into the direct sum of three orthogonal nonzero nonisotropic subspaces. As dim E > 3, at least one of the Pi is of dimension greater than 1. Let x be a nonisotropic vector in Pj where dim Pj > 1. Let w be the 1-involution with subspaces W+ = xK and W = (xK); w is in Un(K,f) be Section 1.6. As W is in Pj, it is left invariant by u and v. Then W = (W )' is left invariant by u and v, and, consequently, u and v commute with w. That is, w is in c(u,v); so any r in c(c(u,v)) must commute with w, as w is a l-involution. Then r(x) = xb and this must hold for every nonisotropic vector x in P.. Consequently by Lemma 1.2, r(x) = x6j with 8j in Z. Furthermore r must commute with u and commute or anticommute with v, as u and v are in c(u,v). However, r cannot anticommute with v, for then would r(P1) = P2 whereas dim P1 / dim P2. So r must leave all Pi invariant. Consequently if dim Pk = 1 and x is in Pk, r(x) = x8x. But r2(x) = xe for all x in E with e in C. Therefore 6 2 = 8 2. But if an element e of a sfield has a square root 6 in the x j center, it has but two square roots +6 inasmuch as the polynomial x2 - c then factors. So 6x = +j6 and 6x is in Z. Consequently, for all

25 vectors y = x0 with 0 in K, r(y) = y6x. Set 6k = 6x and 6 = 63. Then for x in Pi where dim Pi > 0, r(x) = +x6. This gives two choices in sign for r(x) for x in P4 and in P1 or P2, whichever is not zero. So there are but four possibilities for r, each of which is a representative of 1, u, v, or uv. Therefore v(u,v) = 4 In subsequent cases, situations similar to that above arise; and when parts of the proofs are identical with what we have just given, they will not be rendered in such detail. 2.2 Extremal Involution Commuting with an Involution of the Third Kind Let u be an extremal involution and v be an involution of the third kind. Again take u in u to be a 1-involution and v in v such that v2 = P in Cn with = p.2 where p. is not in Z. As dim E > 2, v must commute with u. With regard to v, form the group Un(Kl,fl) acting on the space V+ where K1 is the centralizer of p. in K (cf. Section 1.7). The center Z1 of K1 is then Z(i). The restriction ul of u to V+ is a 1involution in Un(Kl,fi) with eigenspaces U1+ = U+V+V and U1 = U-'V+. For the remainder of this section, we assume merely that ul is the restriction of a projective involution u commuting with v such that ul is of the first kind in Un(Kl,fl). This generalization will be needed in Chapter III. The K1-subspaces U1+ and U1- are orthogonal and nonisotropic (with respect to fl), at least one of which, say U1, is of Kl-dimension greater than 1 as n > 2. Let x be a nonisotropic vector in U1+ and wl be the 1-involution of Un(Kl,fl) with subspaces W+ = xK1 and W1 = (xK1)iL. Then w commutes with ul and its extension w to E is a 1involution of Un(K,f) commuting with u and v. Consequently w is in c(u,v) and r in c(c(u,v)) must commute with it.

26 Now r or hgr = r' commutes with v' = p- v. Since how = who, r' as well as r commutes with w. Then the restriction ri of r or r' to E1 commutes with wl. Therefore rl(W1+) = W1+ and r(x) = x6x. If r commutes with v', then clearly rl is a projective involution of Un(K1,f) such that r2 = c. If r anticommutes with v' then rl is a projective semilinear involution relative to the automorphism @3 of K1 and rl2 = EQ2 where e = r2 and e in Z. But Q was chosen in Section 1.7 so that Q2 = a is in Z. So rl2 = cE is in Z and 9 = 1. In both cases, rl satisfies the hypothesis of Lemma 1.2. Consequently r(x) = x6 with 8 in K1 for all x in U1. Furthermore, since ~ is not an inner automorphism of K1, rl cannot be semilinear with respect to Q. Consequently rl is a projective involution in Un(Kl,fl) and 6 is in Z(,). As r12 = e, 62 = e where c is in Z. But then the theorem on cyclic fields (cf. Section 1.9) yields that 6 is in Z or 8 = 6 oi with 8, in Z. If dim U1- > 1, the same argument shows that r(x) = x- with 6- in Z(,u) for x in Ul-. Again (68)2 = 82; so 68 = +6. If dim U1T = 1, then as in Section 2.1, rl(Ul") = U1I and r(x) = x6b for x in U1-. But 8 2 = 62; so 6x = +6 and is in Z(i.)o This implies that 6x is independent of x in Ul-; so we set 6x = 5-. Since ul is of the first kind in Un(Kifl), u12 = 7 where 7 = X2 and X is in Z(i). On the other hand u2 = Z with 7 in Z. Then by the theorem on cyclic fields, x is in Z or X = \,u with %o in Z. If the former is the case, u is an involution of the first kind and we can always pick u in u to be an involution in Un(K,f). If the latter is the case, we can consider the X = v by replacing the given u by \ lu, which is also in u. Now let 11, ul, v,, and ujvl be

27 the restrictions of 1, u, v, and uv respectively to V+. The identity is 11 and vl is the homothetic mapping hi. If u is an involution in Un(K,f), ul is an involution with subspaces U1+ and U1- and ulvl is a projective involution such that ulvl(x) = +xL for x in U1, respectively. If u is a projective involution of the third kind in Un(K,f), then ul(x)= +xju according as x is in U1+ or U1 and ulvl(x) = +xP with P in C for + x in U1- respectively. As rl is in Un(KK,fl) and U1+ and U1- are nonisotropic, 5Tb = 1. If 6 is in Z, the 8T = bJ and hg is in Cn. If 6 = 8o0, by looking at vl irl, we can similarly conclude that 8 J65 = 1 and h05 is in C. So in any case, there exists an element h3 in Cn such that h r1 coincides with one of 11, ul, vl or ulvl. Then r will always be in one of the cosets 1, u, v, or uv. Consequently, we have v(u,v) = 4. In the case with which we began, this yields THEOREM 2.1 If the dimension of E is greater than 3, and u is an extremal involution in PU (K,f), v- = 4. ~_,,n U 2.3 Nonextremal Involutions of the First Kind Let u be a p-involution with p > 1. We now restrict n to be greater than 4 and n f 6. We shall exhibit an involution v commuting with u such that v(u,v) > 4. Pick u in u to be a p-involution in Un(K,f) with subspaces U+ and U-. Let P1 be a 1-dimensional nonisotropic subspace of U+ and P3 a 2-dimensional nonisotropic subspace of U. Set V+ = P1 P3, V- = (V+), P2 = U+'V' and P4 = U"V —V Then dim P2 = p-l > 0 and dim P4 = n-p-2. Since p < [n/2] and n > 5, we have dim P4 > 0. So P1, P2, Ps3 and P4 are orthogonal nonisotropic subspaces of dimension greater than zero. Since U = P2 @P4, E = P1 0P2 @ P3sP4

28 Let v be the 3-involution with subspaces V+ and V-; then clearly v commutes with u. Let ri (i = 1,2,3,4) be the involution of Un(K,f) with positive eigenspaces Ri = Pi and negative eigenspace Ri = OR. Each of Jfi j u, v, and uv has a positive eigenspace which is a sum of two Pi; so ri is not one of 1, u, v, or uv. We shall show that ri is in c(c(u,v)). First if t in C(u,v) commutes with both u and v, then t leaves invariant + - + U, U, V, and V; consequently t leaves P1, P2, P3, and P4 invariant. So t leaves Ri and Ri invariant and thus must commute with rio On the other hand t cannot anticommute with v because v is a 3-involution and n i 6. If t commutes with v and anticommutes with u, then t(P1) = P3; but this is impossible as dim P1 / dim P3. So therefore ri is in c(c(u,v)). This proves THEOREM 2.2 If the dimension of E is greater than 4 and not 6 and if u is a nonextremal involution of the first kind, v- > 4. In other words, nonextremal involutions of the first kind are not in 9.

CHAPTER III INVOLUTIONS OF THE THIRD KIND7 In this chapter u will represent an involution of the third kind. We pick a representative u in u. Then u2 = y in Cn and y = k2 where X is not a square in Z. Form the group IUn(Kl,fl) acting on the space E1 = U+ as in Section 1.7. Then if v is a projective involution in Un(K,f) commuting with u, its restriction vl is a projective involution in Un(Klfl) such that v12 = P with D in C. If v is a projective involution in Un(K,f) anticommuting with u, then v is represented by a projective semilinear involution in rUn(Klifl) relative to the automorphism G of K1. Furthermore, since Q2 = a is in Z, ~2 = 1 and vl2 = where a is in Z. We shall consider first the case that v commutes with u. We distinguish four cases in accordance as to whether vl is of the first, second, third, or fourth kind in Un(Klfl). We have already dealt with the case the vl is of the first kind in Un(Kl,fl) in Section 2.2. There we proved the following result. PROPOSITION 5.1. Let v be a projective involution in Un(K,f) commuting with u such that its restriction vl to E1 is a projective involution of the first kind in Un(Kl,fi). If n > 3, then v(u,v) = 4. We shall show that this result holds in the three other cases although with different restrictions on n. TIt is not necessary to determine whether involutions of the second kind are in 9. The argument in Chapter V allows us to pass over this case. 29

30 3.1 Commuting Projective Involution whose Restriction is of the Second Kind Let vl be of the second kind in Un(Kl,fl). Assume n > 4. Then v12= in Cn, P =,J2 with pt in Z(x), and 1T31 = -1. Also E1 = Vi+ V1where Vi+ and V,- are totally isotropic subspaces of E1 such that vl(x) = +xL for x in V1+, respectively. Furthermore, dim V1+ = dim V1- = n/2 > 2. Let x be a vector in V1+; then there exists a vector y in V'1 such that fl(x,y) j 0. Let W+ be the subspace of E1 generated by x and y. Since x and y are nonorthogonal isotropic vectors, W+ is nonisotropic (cf. Section 1.4). Set Wi" = (Wi+)L and let wl be the 2-involution in Un(Kl,fl) with subspaces W+ and W1I. As W2+ = xK1 ~yK1, vl leaves W1+ invariant. As vl is in Un(Kl,fl), vl leaves Wi = (W1+YL invariant. Consequently wl commutes with vl and the extension w of wl to E is a 2-involution commuting with u and v. The choice of the vector y in V1 is not unique. Indeed, let y" be a vector in Vi W1- and set y' = y+y". Then fl(x,y') = fl(x,y) i 0 and y' is linearly independent of y. Set W1'+ = xK1 ~y'K1 and Wi'- = (Wi'+).L. Then if wl' is the 2-involution of Un(Kl,fl) with Wi and W1' as its subspaces, wl' commutes with vl, and its extension w' to E commutes with u and v. Not let r be in c(c(u,v)). As n > 4, r commutes with w and w'. If r anticommutes with u, we form r' = hQr; r' also commutes with w and w'. The restriction rl of r, in case r commutes with u, or of r', in case r anticommutes with u, then commutes with wl and wl'. This means that rl(W1+, W'+) = Wl+W1'+; that is, ri(x) = x5x. Furthermore, r since rl is linear or semilinear ) = with respect to and r with in Z, we have by Lemma 1.5 that rl(x) = x5 with 5 in K1 for all x in

31 V1'. Furthermore, the possibility that rl is a semilinear transformation of E1 is eliminated since @ is not an inner automorphism of K1. So rl must be linear and 8 is in Z(x). Similarly, for x in V1', rl(x) = x8' with 68 in Z(\). Now r12 = E in Z. This implies that 82 = (8)2 = e. Hence, 5- = +8; and by the theorem on cyclic fields (cf. Section 1.9), 8 is in Z or 8 = 8oX with 50 in Z. If 8 is in Z, take x in V1+ and y in V1- such that fl(x,y) t 0. Then fl(x,y) = fl(rl(x),rl(y)) = +8T8fl(x,y). So 6T8 = +1 according as 8 = +8. As 8 is in Z, ST = J. So we have that 5 is in C in case 5- = 8. If 8' = -8, then the above argument applied to vlrl shows that vlrl(x) = x8tJ for all x in E1 where 581 is in C. If 6 = 58k with 68 in Z, replace ri by ulrl is the above arguments. We then obtain that ulrl(x) = x607 with 60y in C or vlulrl(x) = xbol'7 with 60o7 in C. In any case, there exists a homothetic transformation h1 in Cn such that Nrl coincides with the restrictions of 1, u, v, or uv to E1. Hence r is in 1, u, v, or uv. This yields the following result. PROPOSITION 3.2 Let v be a projective involution in Un(K,f) commuting with u such that its restriction vi to E1 is a projective involution of the second kind in Un(Kif1). If n > 4, v(u,7) = 4. 3.2 Commuting Projective Involution whose Restriction is of the Third Kind Let vl be of the third kind in Un(Ki,fl). Assume n > 2. Then vj= - with P in C and P = p2 where p. is not in Z(x). Clearly, then v is the restriction of an involution of the third kind in Un(Ki,fl). Then apply the analysis of Section 1.7 to va. Let K2 be

32 the subsfield of Kz left elementwise invariant by pi. Then K1 = K2 @ K2f where ~ = -i and *2 is in Z(x). The center of K2 is Z(\,p). Also E1 as a right vector space E1* over K2 is the direct sum of the eigenspaces of vi: El* = V1+ (T)V1-. Set E2 = V1+ and form the group JUn(K2,f2) acting on E2 as in Section 1.7. Then an element t of Un(K,f) which commutes with both u and v can be restricted to an element t2 in Un(K2,f2). Conversely each such element t2 can be extended first to an element tl in Un(Klifl) commuting with vl and then to an element t of Un(K,f) commuting with u and v. A projective involution t of Un(K,f) which anticommutes with u is represented by a projective semilinear involution in Un(Kl,fl) relative to the automorphism Q of K1. In order to restrict or represent tl inrUn(K2,f2) as in Section 1.7, the relation p. = p. must be satisfied. In general, this will not be the case; but we shall show that there exists an element Xin K such that XK = -Y, L = p1, and X2 is in Z(p.). This will enable us to carry out the analysis of projective involutions which anticommute with u or v. First note that kt = x as 4 is in K1 and X is in Z1 = Z(%). We now interchange the roles of u and v to obtain the same relation between X and p. Clearly, we have that. is in K but not in Z. So let K1' be the fixed subsfield of K by u1 = pL. Then K = K1' (Kl' where f is the element chosen above. Since pi is in K1, ix = X.; so A is in K1'. Clearly, X is not in the center Z(p.) of K1'; for, otherwise pL would be in Z() So let K2' be the fixed subsfield of K1' by ~. Then K1' = K2' G K2'X where now X is chosen as in Section 1.7 so that )t = 4_ andS2 is in Z(p,). We cannot say that(2 is in Z as is the

33 case with 9, but this will not hinder us. To summarize, we have the following relations: i) K = K2 ~ K2*j e K27 e K2*X; ii) )= -k; A =; = k; A = A; iii) X2 is in Z(t); *2 is in Z(X). We now replace 9 by\. This does not alter the situation with respect to the projective involutions commuting with u or v. Let t be a projective involution in Un(K,f) which anticomrutes with u. Then t is represented by a projective involution t, inrUn(Kl,fl) which is semilinear with respect to the involutive automorphism S of K1. Furthermore, if t commutes or anticommutes with v, t1 commutes or anticommutes with vl. Then by Section 1.7, tl can be restricted to a semilinear projective involution in rUn(K2,f2) relative to the automorphism 8 of K2 in case ti commutes with vl or relative to the automorphism X*r of K2 in case tl anticommutes with v,. If t commutes with u and anticommutes with v, then the restriction tl of t to E1 anticommutes with vl and hence is represented by a projective semilinear involution t2 relative to the automorphism i of K2. The extension w of a 1-involution w2 of Un(K2,f2) to E is a 1-involution in c(uv)o Consequently, a projective involution r of c(c(u,v)) commutes with w as n > 2. Therefore, if r2 represents r in rUn(K2,f2), r2 commutes with w2. This means that r2 leaves every 1-dimensional nonisotropic space of E2 invariant. Furthermore, r2 is a projective semilinear involution relative to an involution automorphism of K2 and r22 = e with e in Z(X,p.). So by Lemma 1.2, r2(x) = x6 with 8 in K2 for all x in E2. Moreover, r2 must be semilinear with respect to an inner automorphism of K2. Since 5

r and \r are not inner automorphisms, we have that r2 must be linear and e is in Z. Furthermore, 5 is in Z(X,j)) and 62 = e. We now apply the theorem on cyclic fields twice. First either 8 is in Z(<) of 6 = 816j with 61 in Z(k). Then & or 81 is a square of an element in Z. This implies that 8 is Z, 6 = &61j, 6 = 82X, or = 834l with 8i in Z. Set 60 = 6. Let u2 and v2 be the restrictions of u and v to E2. Let r2' be that one of r2, u2 lr2, V2-lr2Por U2-1v2lr2 such that r2'(x) = xbi for x in E2 and some i = 1,2,3,4. Since r2 is in Un(K2,f2), 8 is in Z, and E2 is nonisotropic, ST8 = = 1. Therefore,the extension r' of r2' to E is in Cn and r is in one of the cosets 1, u, v, or UV. This gives us the following result. PROPOSITION 3.3 Let v be a projective involution in Un(K,f) commuting with u such that its restriction v, to E, is a projective involution of the third kind in Un(Klfl). If n > 2, v(u,6 ) = 4. 35.. Commuting Projective Involution whose Restriction is of the Fourth Kind Let vl be of the fourth kind in Un(Klfl). Assume n > 4. Then vl2 = P with P in C and P not a square in K1. As in Section 1.8, extend the sfield K1 to a sfield Li by adjoining a square root p of P. The center L1 is then Z(X,p). Then E1 can be considered as a right vector space over L1 by setting xp = v1(x) and f2 can be extended to a nondegenerate hermitian sesquilinear form g, defined on F1. We then form the group PUm(Li,gl) acting on F1. Let tl be the restriction of a projective involution in Un(K,f) which commutes with u or the representation of a projective

involution which anticommutes with u, in which case tl is semilinear with respect to the automorphism @ of K1. If, furthermore, t commutes or anticommutes with v, tl commutes or anticommutes, respectively, with vl. Then tl can be considered as a projective semilinear involution in rUm(Ll,gz) relative to one of the automorphisms 1, @, a or ga where a is the automorphism of L1 defined by a: p + -p. It is easily verified that a and g commutes; so all these automorphisms are involutive. The projective involutions u, v, and uv are represented in UM(Ll,gl) by the homothetic transformations h., hp, and hep, respectively. Every 1-involution wl of Um(Ll,gl) extends to a 2-involution w of Un(K,f) in c(u,v). If rl in PUm(Ll,gl) represents a projective involution r in c(c(u,v)), thenr must commute with w as n > 2 and ri must commute with wl. So rl leaves invariant every 1-dimensional nonisotropic subspaces of F1. Furthermore, rl is a projective semilinear transformation relative to an involutive automorphism of L1 and r2 = e with E in Z(x). So by Lemma 1.2, rl(x) = xb with 6 in L1 for all x in Fl. However, rl cannot be semilinear with respect to the automorphisms, Q, a or Ga as none of these are inner automorphisms. Consequently, rl is linear and must be the representation of a projective involution r commuting with u and v. Therefore, is in Z and 6 is in Z(X,p). Also 2 = c. So applying the theorem on cyclic fields twice as in Section 3.2, we have that 6 = 60, 8 = 61p, 8 = 82k or 6 = 83Xp where 81 is in Z and biJbi = 1. Thus rl is the representation of a projective involution in 1, u, v, or uv. This yields the following result.

36 PROPOSITION 3.4 Let v be a projective involution in Un(K,f) commuting with u such that its restriction vl to E1 is a projective involution of the fourth kind in Un(Kl,fl). If n > 4, v(u,v) = 4. Now we must consider the case that v anticommutes with u. Then set v' = hev. Now v' is a semilinear projective involution in PUn(K,f) which commutes with u' = X'lu. So the restriction v2 of v' to E1 is a semilinear projective involution in rUn(Ki,fi) relative to the automorphism 9 and v12 = B with P in Z. First note that if vl(x)=xJ' V O for x in E1, then vl2(x) = x>>. Therefore P = Q10. So now we distinguish between two cases: Case I where such an element,u of K1 exists and P is of the form,11Q and Case II where no such element exists in K1 and P is not of the form uig. 3.4 Anticommuting Projective Involutions. Case I Let v be a projective involution in Un(K,f) anticommuting with u such that v is represented by a projective semilinear involution vl in rUn(Kl,fl) where vl2 - P and P =,pQ with pt in K1. Assume n > 2. Then let K2 be the fixed subfield of K1 by QJ. Since QL = -%, K1 = K2 GK2X. Let Z2 be the center of K2; then an element in Z2 commutes not only with every element of K2 but also with X and therefore with every element of K1. So Z2 C K2,^Z(k) = Z. Since Z is left elementwise invariant by Qi, Z C Z2. Therefore Z = Z2. In the manner of Section 1.7, form the group FUn(K2,f2) acting on E2 = V+. Now an element t of c(u,v) which commutes with u and v commutes also with hQv and so can be restricted to an element tl of rUn(Kl,fl) which commutes with vi. Then tl can be restricted to

37 an element t2 of rUn(K2,f2). Since t is linear in Un(K,f), tl will be linear in Un(K2,f2). If an element t commutes with u and anticommutes with v, then ut will commute with both u and hev and hence ut can be restricted to a linear transformation in Un(K2,f2). Similarly, if t anticommutes with u and commutes with v, form vt; if t anticommutes with both u and v, form uvt. Every 1-involution w2 of Un(K2,f2) can be extended to a 1involution w of Un(Kf) which commutes with u and v. Hence r is in c(c(u,v))andmust commute withw. Denote by r' that one of r, ur, vr, or uvr which commutes with u and v. Then r' will still commute with w and its restriction r2 will be a linear projective involution of Un(K2,f2) which leaves every 1-dimensional nonisotropic subspace of E2 invariant. Hence by Lemma 1.2, r2(x) = x8 with 8 in Z for every vector x in E2. Furthermore, 6H = 1 where H is the anti-automorphism corH 8Jo responding to f2. But since 6 is in Z, 6. So r2 can be extended to a homothetic transformation of E. That is, r' is in 1; so r is in one of 1, u, v, or uv. This yields the following result. PROPOSITION 5.5. Let v be a projective involution of Un(K,f) anticommuting with u such that v is represented by a projective semi. linear involution vl of rUn(Kl,fl) where vI2 = l and P = XX. If n > 2, v(u,v) = 4. 3.5 Anticommuting Projective Involutions. Case II Now consider the case that v anticommutes with u and is represented by a projective semilinear involution vl acting on E1 such that v12 = P where P is not of the form tL,~ for in K1. We then adjoin a square root p of D to K1 forming a sfield L1 = Kl(p) and

38 form the group rUm(Ll,gl) acting on the space F1 as in Section 1.8. Now p satisfies the relation Op = p~Q for all 0 in K1. In particular if 0 is in Z1, the center of K1, then ~0 = 0. On the other hand 0( + pT) = (6 + pT)0 for all S and T in K1. This implies that - 0 = p(rl - Ar). So 0 must commute with t and hence Z1 C Z(X). This implies that Z1 = Z. Again an element t in Un(K,f) which commutes with u and v commutes with hev and hence can be restricted to a projective involution tl in Un(Kl,fl) which commutes with vl. Hence ti can be considered to be in Um(Ll,gl). If t anticommutes with one or both of u, v, or uv, then one of ut, vt, or uvt will commute with both u and v. Now let wl be an 1-involution of Um(Ll,g1). Then wl can be extended to a 2-involution w in Un(K,f) which commutes with u and v. Since n > 4, r in c(c(u,v)) must commute with w. Denote by r' that one of r, ur, vr, or uvr which commutes with u and v. Then r' commutes with w and is represented by a linear projective involution rl' of rUm(Ll,gl) commuting with wl. Hence rl' leaves invariant every 1-dimensional nonisotropic subspace of Fl. Therefore, by Lemma 1.2, rl'(x) = x8 with 8 in Z for all x in Fi. Again as in Section 3.4, this implies that s8J = 1 and thus that r' is in 1. Hence r is in one of 1, u, v, or uv. This yields the following result. PROPOSITION 3.6 Let v be a projective involution of Un(K,f) anticommuting with u such that v is represented by a

39 projective semilinear involution vl of frUn(Kl^fi) where vl2 = 3 is not of the form [pAl for, in KI. If n > 4, v(uv) = 4. Now Propositions 3.1 through 3.6 yield the following theorem. THEOREM 3.1. If the dimension of E is geater than 4, and if u is an involution of the third kind in PUn(Kf), then v- = 4. So all involutions of the third kind are in n.

CHAPTER IV INVOLUTIONS OF THE FOURTH KIND In this chapter, u will represent an involution of the fourth kind in PUn(Kf). Pick a representative u in T. Then u is characterized by the relation u2 = 7 in Cn where 7 is not a square in K. As in Section 1.8, we form the sfield L obtained by adjoining a square root p of 7 to K. The center of L is Z(p). Then E can be considered as a right vector space F over L. Form the group rUm(Lg). Let a be the automorphism of L defined by a: p + -p. An element t of Un(Kf) which commutes or anticommutes with t can be considered as an element of pUm(L,g) which is, respectively, linear or semilinear with respect to the automorphism a. Throughout this chapter, we assume that n > 4. We shall show that vY = 4. Let v be a projective involution of Un(K,f) which commutes or anticommutes with u. We consider v to be in rUm(L,g) and distinguish various cases. Since u is in c(u,v), this set can be considered to be a set of projective semilinear involutions in FUm(L,g) which are linear or semilinear with respect to the automorphism a of L. Similarly, the set c(c(u,u) can be considered to be in rUm(Lg). 4.1 Projective Involutions of the First Kind in Um(L,g) Let v be a projective involution of the first kind in Um(L,g). Then v2 = P with P in C, P =,2 where L is in Z(p) and JiL = 1. As in Section 1.5, F = V 0 V where Vt and V are the orthogonal nonisotropic eigenspaces. Since m > 2, one of V+and V-, say V^, must be 40

of L-dimension greater than 1* Let x be a nonisotropic vector in V+ and w the 1-involution of Um(Lg) with xK as its 1-dimensional subspace W+. Since W+C V+ and v is in Um(L,g), v leaves invariant W+ and W = (W+)1. Consequently, w commutes with v, and thus, w is in c(u,v). Let r be in c(c(u,v)). Then as an element of PUm(L,g), r is linear or semilinear with respect to the automorphism a. Furthermore, r must commute with w as m > 2; hence r(x) = x8x with 8x in L for each nonisotropic vector x in V+. So by Lemma 1.2, r(x) = x8 with 5 in L for x in V+. Also, r must be semilinear with respect to an inner automorphism of L. But a does not leave the center Z(p) of L elementwise invariant and, therefore, is not an inner automorphism of L. So r is linear and 5 is in Z(p)o Also r2 = e with e in Z; so 52= e. It then follows from the theorem on cyclic fields that 8 is in Z or - = 50p with 6o in Z. If dimL V- > 1, it follows in a similar manner that r(x) = x6-with &' in L for x in V; and as r2 = E, we have (5-)2 = 82 = e. So 8- = + 5. If dimL V- = 1, then r(x) = x6x for x in V-. Again, 8X2 = 52 with 5x = + 8, So 5x is in Z(p) and hence is independent of x in V-. Set 8x = 5. Since g is sesquilinear with respect to the anti-automorphism J extended to L, and V+ is nonisotropic, we have that 8b = 1. Thus if 8 is in Z, 5 is in C. If 8 = 80o, 6OJ8o = 1 as pJ = p-l; so o6 is in C. In any case r as an element of Un(K,f) is in one of the cosets 1, u, v, or w. So we have proved the following result. PROPOSITION 4.1 Let v be a projective involution of the first kind in Um(L,g). If n > 4, v(^V) = 4.

4.2 Projective Involutions of the Second Kind in Um(L,g) Let v be a projective involution of the second kind in Um(L,g). Then v2 = 5 with D in C, P = w2 with k in Z(p), and [Jk = -1. As in Section 1.6, F = V+ V V where V+ and V are totally isotropic subspaces of L-dimension m/2. Now let x be an arbitrary vector in V+ and pick y in Vsuch that g(x,y) i 0. Then W+ = xL ~ yL is nonisotropic and the 2-involution w in Um(L,g) with W+ and W- = (W')L as eigenspaces commutes with v. As w is in Um(Lg), it can also be considered to be a 4-involution in Un(K,f) which commutes with u; that is, w is in c(uv). Also W- = W-rV+f e W-V'V. Both W-nV+ and W-nV- are totally isotropic; andas W~ is a nonisotropic subspace of L-dimension m/2,dimL W-rV+ = dimL W>nV-= m/2 - 1. Since m > 2, these subspaces are not zero and so there exists a vector b in W-^V-. Set y' = y+b; then g(x,yT) = g(x,y) i O. Consequently, the involution w' with subspaces W'+ = xL @ y'L and W' = (W'+)k is also in c(u,v) when considered as a transformation in Un(K,f). First consider the case that r in c(c(u,v)) commutes with w. We know that this must happen when n i 8. Then we can conclude that r commutes with w'. For, should r anticommute with w', then r(x) would not be in V+ rW+ = W+W'+ and so would be in V-rW, as r must commute or anticommute with v. We can then take y in V-nW+ such that r(x) = y and y would be in W'- and hence orthogonal to x in W'+, which is a contradiction. Consequently, r commutes with both w and w' and r(x) is in W+ W'+; that is, r(x) = xBx with 5 in L. Now consider the possibility that r commutes with v but anticommutes with w. Then n = 8 and r(x) = a with a in V+rW-. Pick b in V-nW- as above. Then dimL V+IW- = dimL V-nW = 1; so a and b generate W, which is nonisotropic. Consequently g(ab) # O Furthermore,

43 r(x) = a is not in W'+ = xK y'K as V+nW'+ = xK; therefore, r must anticommute with w' and a must be in V+TW'. But a is not orthogonal to y' = y+b,as g(a,y') = g(a,b) i 0 while y' is in W'+. Thus a cannot be in W'I; and, consequently, r cannot anticommute with w'. But as w' is in c(u,v) and r is in c(c(u,v)), this is a contradiction. Finally consider the possibility that r anticommutes with both v and w. Again n = 8 and r(x) = b with b in V-nW-. Since r(V+n^W) = V-fnW+, there exists a vector a in V+rW- such that r(a) = y. Set z = x+a and, as before, y' = y+b; then r(z) = y'. On the other hand, r2 =; so r(y) = ac and r(b) = r2(x) = xE. From this it follows that g(z,y') = g(x+a,y+b) = g(x,y) + g(a,b) = g(x,y) + g(y,x)e. Since y was chosen arbitrarily to form W+, it may be replaced by y0 with 0 in L. Then there always exists a vector y' such that g(z,y') i 0. Indeed, if g(x,y)0 + 0 g(y,x)E = 0 for all 0 in L, we obtain upon setting g(x,y) = i ~ 0, 0 + 0Je = 0. Setting 0 = 1, we have -J = -e- 1 and then that J = 0t5-1. Thus K must be commutative as J is an anti-automorphism, but then J must leave Z(p) = K elementwise invariant. However, even in this case where U8(K,f) has to be the Orthogonal group 08(K,f), J, which is the identity on K, extends to L = K(p) = K(i) as the automorphism J: i + -i (cf. Section 1.8). This is in contradiction due to the assumption g(z,y') = 0 for all 0 in K. So we can take g(z,y') # 0. Then the subspace Q+ generated by z and y' must be nonisotropic and r must commute with the 2-involution q with Q as its positive subspace. But now we are reduced to the case at the beginning of this discussion where we had r commuting with w instead of q as

44 here. The argument there tells us that r(z) = z8z; but r(z) = y'. So we have a contradiction and r must commute with w and w'. That is, r(x) = xbx with 5x in L. Now by Lemma 1.3, it follows that r(x) = x5 with 6 in L for all x in V+. Furthermore, r cannot be semilinear with respect to a as a is not an inner automorphism of L for the reason that it does not leave Z(p) elementwise invariant. Consequently, r is linear and 5 is in Z(p). Furthermore, 82 = e in Z. So by the theorem on cyclic fields (Section 1.9), 8 is in Z or 5 = 85o with 5o in Z. A similar argument shows that r(x) = x6- with 8- in Z(p) for all x in V-. Since r2 = c, it follows that 8- = + 8. But now if 8 is in Z and 8- = S, r is the homothetic mapping h8 on F. Since F is nonisotropic and r is in Um(L,g), 85 = 1; that is, r is in 1. If 8 is in Z and 5 = -8, then vr is a homothetic mapping h 8 on F; for, v(r(x)) = v(x)8 = xi8 if x is in V+ and v(r)x)) = -v(x)8 = x68 if x is in V-. So vr is in 1 and r is in V. Similarly if 8 = o0p and 8- = 5, ur is a homothetic mapping in Cn and r is in u; and if 8 = 80p and 8- = -8, uvr is a homothetic mapping in Cn and r is in uv. So r must be one of 1, u, v, or uv. Therefore we have the following result. PROPOSITION 4.2 Let v be a projective involution of the second kind in Um(L,g). If n > 4, v(u,v) = 4. 4.3 Projective Involutions of the Third Kind in Um(L,g) Let v be a projective involution of the third kind in Um(L,g). Then v2 = 5, B = p2 where i is in L, but not in Z(p). As in Section 1*7, let L1 be the fixed subsfield of K1 by L and let 0 be an element of L such that 0 = -0 and g2 is in Z(p). Then L = L1~ L10. The center of L1 is

Z(p^,). Then F can be considered as a right vector space F* over L1, and F* = V+ i V. Set F1 = V+, and take the form g1 as in Section 1.7. Then form the group rUm(Ll,gi) acting on the space F1. If t is a projective involution of Um(L,g) which commutes with v, then by Section 1.7, t can be restricted to a projective involution t, of Um(Li,gi). Conversely, if tl is a projective involution of Um(L1,gl), t1 can be extended to a projective involution t of Um(L,g)& If t anticommutes with v, then t is represented by a projective semilinear involution tl of rUm(Llgl) which is semilinear relative to the automorphism 9 of L1. The situation is not quite so simple if t is a projective semilinear involution of pUm(Lg) relative to the automorphism a. If we can show that 2a = then by Section 1.7, we can restrict t to a semilinear projective involution tl of FUm(Lilg) which is semilinear with respect to the automorphism a. Indeed, as we shall see, this is true if t is in c(c(u,v)) and t commutes with v. Furthermore, if t is such a projective semilinear involution in c(c(uv)) which anticommutes with v, then ia = -pL; but then we show that t has a restriction t1 to Fl which is a projective semilinear involution of rUm(Llgl) relative to the automorphism a of L. Therefore, let t be a projective semilinear involution of FUM(Ll,gl) relative to the automorphism a of L which is in c(c(u,v)). Let x be a vector of F1 which is nonisotropic with respect to the form gl. Set Wz = xL1 and W1 = (W+)l, and let wl be the 1-involution of Um(La,gl) with W1 and W1 as its eigenspacest As m > 2, a projective semilinear involution r in c(c(u,v)) must commute with the extension w of w, to F. In particular t commutes with w. As a transformation

46 of F, w has positive eigenspace W+ = xL1 0 xL1O = xL. Therefore r(x) = x6x with 8x in L. Set 6x = 5. Since r2 = e in Z, we have 86 = e. Then it is easily verified that 8a = a8; so the automorphism X = &a is an involution. Let Lo be the fixed subsfield of L by A. Then Lo is of index 2 in L. Furthermore, pt = p5a = pa = -p So L = Lo 0 Lop* We next show that Lo = K. Let 0 be in K; then 0 = 01 + p02 where 01 and 02 are in Lo. But 0_ = 0; so 01a - p02a = 01 + P02. This implies that 01 = 01 and 02a = -02. So 0 is in K and 02 = P03 with 03 in K. Thus 0 = 01 + 037 = P(01P- P+0) = = 0 + P02, and the representation of 0 is unique only if 02 = 0 and Lo = K or if 01 = 0 and Lo = Kp. The latter case is excluded as Kp is not a field. Therefore Lo = K. But then the restriction of x = b5 to Lo = K coincides with the restriction of 8 to Loo But also X is the identity on Lo; so 8 commutes with every element of K. This means that 8 is in Z(p). Indeed, if 8 = 81 + 82P with 81 and 82 in K, 0 = 80 for 0 in K implies that 061 = 810 and 082 = 820. Hence 81 and 82 are in Z and 8 is in Z(p). But then t(v(x)) = t(x)l = x81l and v(t(x)) = v(x)8 = x8b. So if t commutes with v, Ha = p. and the method of Section 1.7 applies and-. we can restrict t to F1. If t anticommutes with v, then Ad = -p.. Therefore, pia = aS and a leave L1 invariant. Furthermore, t also anticommutes with hi-l; so t commutes with v' = h._lv and consequently can be restricted to a semilinear projective involution of rUm(Ll,gl) acting on F2 relative to the automorphism a of L1. So in general, a projective semilinear involution r in c(c(u,v)) can be restricted to or is represented by a projective semilinear involution rl in rUM(Li,gl) which is linear or semilinear with respect to the

47 involutive automorphism Q or a of L1. Since the choice of a nonisotropic vector x in F1 was arbitrary, rl must commute with every 1-involution in Um(Llgl). Hence r1 leaves invariant every 1-dimensional nonisotropic subspace of F1. Since r12 = E with E in Z(p,4), Lemma 1o2 yields that r(x) = x8 with 6 in L1 for all x in F1. Furthermore, ri must be semilinear with respect to an inner automorphism of L1. However, neither O nor a is an inner automorphism. Consequently, r1 must be linear; 5 is in Z(p,p); and 82 = e with E in Z. Then, as it was argued in Section 3.3, r is in one of 1, u, v, or uv. This yields the following result. PROPOSITION 4-3 Let v be a projective involution of the third kind in Um(L,g). If n > 4, v(iT,v) = 4. 4.4 Projective Involutions of the Fourth Kind in Um(L,g) Let v be a projective involution of the fourth kind in Um(L,g). Then v2 = f in Cn and D is not a square in L. We adjoin a square root p' of P to L forming the sfield M = L(p') = K(p,p'). The center of L is Z(p,p'). As in Section 1.8, the space F can be considered as a right vector space G over M of dimension q = m/2 = n/4. The form g can be extended to a nondegenerate hermitian sesquilinear form h defined on G relative to the extension of the anti-automorphism J to M. We then form the group rUq(M,h)o An element in Un(K,f) which commutes with u and v can be considered to be in Uq(M,h) and conversely. An element in Un(K,f) which anticommutes with u and commutes with v, which commutes with u and anticommutes with v, or which anticommutes with both u and v can be considered to be in rUq(Mh) relative to the automorphism a, a', or aa', respectively, where a: p -- -p and a': p' + -pl. Now let w be a 1-involution in Uq(M,h); w can be considered as a 4-involution in Un(K,f) which commutes with both u and v. If r is in

48 c(c(u,v)) then r commutes or anticommutes with w. If n > 8, we can conclude immediately that r commutes with w, Let n = 8 and assume r anticommutes with w; then r, as an element of rUq(M,h), is such that r(x) = y where x is in W+, the positive eigenspace of w, and y is in W, the negative eigenspace of w and y is orthogonal to x and is nonisotropic as h(y,y) = h(x,x) # 0. Also r2 = e with e in Z, so r(y) = r2(x) = xc. By Lemma 1.1, for every element 0 of M, there exists an element t = 0 0 + 1, or 0 - 1 such that zg = x+yr is nonisotropic; and r commutes or anticommutes with the 1-involution wi in U (Mh) with z4M as its positive eigenspaceo In the first case r(z*) m zS6&. This yields y + xll = x8+ + y68r. Consequently, 56 = f-1 and, as 6*2 = e, *-2 = c. If r anticommutes with w*, then r(x*) = r(x+yf) = xfe + y is orthogonal to z*. Then h(z*, r(zf)) = *ce+*Jr = 0 where e = h(x,x) # O0 This yields that But suppose now that 0 m p + p';'-and:choose f = p + p' + k where k = 0, + 1 in accordance with Lemma 1.1. In any event, the square of f is not in Z and hence r-2 # ~; so r does not commute with w1 in this case. Also j-11-*J~ = tzlJ 1 -6. For, suppose that *E = -*J. By Section 1.8, pJ = p-1 and p'J = p-. So we have -(p-i + p'-' + k) = pE + p's + ke. Multiplying by p, we obtain -1 = De + pp'E + pp'- + kp(e+l), which is a contradiction. So r can neither commute nor anticommute with w., which is in c(u,v). Since r is in c(c(u,v)), this is a contradiction to the assumption that r anticommutes with w in the case n = 8, This means that r commutes with every 1-involution in Uq(M,h) and hence leaves every l-dimensional nonisotropic subspace of G invariant. Then by Lemma 1.2, r(x) = x& with 5 in M for all x in G. If r is linear,

then 5 is in Z(p,p'). If r is semilinear with respect to an automorphism it = a, a', or aa' of M, then lto be an inner automorphism, must leave Z(p,p') elementwise invariant, which it does not. Hence 8 is in Z(p,p') and S2 = e in Z. Again applying the theorem on cyclic fields twice, we obtain that 5 = So,, = 81P S = 82P', or 6 = b3pp' with Si in Z. As r is a homothetic mapping in Uq(Mh), it follows that &J& = 1 and hence that 5iJbi = 1. Then one of r, ur, vr, or uvr is a homothetic transformation hq in Uq(M,h) where T is in C. So hq can be considered in Cn and r is in one of the cosets 1, u, v, or uv. Therefore, we have the following result. PROPOSITION 4,4 Let v be a projective involution of the fourth kind in Um(L,g). If n > 4, v(u,v) = 4. Now we treat the case that v considered as a projective involution of Un(K,f) anticommutes with u. Then v considered as an element of rUM(L,g) is a projective semilinear involution relative to the automorphism a of L. If v2 = f where P is in C and if there exists a vector x in F such that v(x) = xp., then v2(x) = X^M.f. We therefore distinguish the two cases: P is of the form,wia with -+ in L (Case I) and P is not of the form,uLa with ji in L (Case II). 4.5 Projective Semilinear Involutions in rUm(L,g). Case I Let v be a projective semilinear involution in rUm(L,g) relative to the automorphism a of L such that v2 = P where P = pLL with i in L. Then as in Section 1.7, form the subsfield L1 of L consisting of elements of L left invariant by the automorphism a' = -la. Since a'2 = 1, L1 is of degree 2 in L and L = L1, L1Q where

50 Q is an arbitrary element in L such that 0a = -0. In particular we can choose p = 0. Then every element 0 of Z1, the center of L1, commutes with p and every element of L,. So if ~ + pTl is in L with ~ and ri in L1, 0(5 + pn) = (~ + pT)0 and 0 is in Z(p), the center of L. Furthermore, a' leaves Z elementwise invariant so ZC Z1 C Z(p)nL,. But p is not in L1, so Z = Z1. Now F when considered as an L1 vector space decomposes into the direct sum F = V+ G V'. Set F1 = V+ and form the group rUm(Ll,gj) as in Section 1.7. An element t of Un(K,f) which commutes with both u and v can be restricted to be in Um(Ll,gl) and conversely. If t anticommutes with u and commutes with v, then vt will commute with both u and v and can be restricted to Um(Llgl). Finally if t anticommutes with both u and v, then uvt commutes with both u and v and can be restricted to UM(Ljgj). Let wz be a 1-involution in Um(Ll,gj), then wl can be extended to a 2-involution w in Un(K,f) which commutes with both u and v. Hence r in c(c(u,v)) will commute with w. Let r' be that one of r, ur, vr, or uvr which commutes with u and v and let rl be the restriction of r' to F1. Then r' commutes with w; so rl commutes with wl. Thus rl is in Um(Ll,g1) and r1 leaves invariant every 1-dimensional nonisotropic subspace of F1. Consequently, by Lemma 1.2, rl(x) = x& with 5 in Z for all x in Fl. Furthermore, sT8 = 1 where T is the anti-automorphism of L1 corresponding to g1. Since 5 is in Z, 5T = &J; so b is in C and the extension r' of rl to E is in Cn. Thus r is in one of 1, u, v, or uv. PROPOSITION 4.6 Let v be a projective semilinear involution in U(L,g) relative to the automohism a of L such that 2 s of the = form P = p4C1 with Ip in L. If n > 4, then v(U,V) = 4.

51 4.6 Projective Semilinear Involution in rUm(L,g). Case TII Let v be a projective semilinear involution of rUm(Lg) relative to the automorphism a of L such that v2 = P where P is not of the form B = 4C with 4 in L. Then, as in Section 1*8, adjoin a square root p' of B to L forming the sfield M = L(p') = K(p,p'). If 0 is in L, the relation 0 p = p'1 holds; in particular pp' = -p'p. Let ZM be the center of M. Then clearly Z C ZMC Z(p) as Z(p) is the center of L. Furthermore p is not in ZM; and since [Z(p):Z] = 2, ZM = Z. Now form the M-space G by defining xp' = v(x) and extend the antiautomorphism J to M by defining p' = p'-. Then, as in Section 1.8, extend the form g to a hermitian sesquilinear form h defined on G and form the group rUq(M,h) where q = m/2 = n/4. An element t of Un(K,f) which commutes with both u and v can be considered to be in Uq(Mh) and conversely. If an element t of Un(K,f) anticommutes with u but commutes with v, then vt commutes with both u and v and so can be considered in Uq(M,h). If t commutes with u and anticommutes with v, then ut commutes with both u and v and so can be restricted to an element in Uq(M,h). If t anticommutes with u and commutes with v, then ut can be restricted to an element in Uq(Mh). If t anticommutes with both u and v, then uvt can be restricted to an element of Uq(M,h). A 1-involution wo0 in Uq(M,h) can then be extended to a 4-involution in Un(K,f) which commutes with u and v. Then r in c(c(u,v)) commutes or anticommutes with w. If n > 8, it follows immediately that r must commute with w. If n = 8, the argument of Section 4.5 shows that r must commute with w. Let r' be that one of r, ur, vr, or uvr which commutes with both u and v. Then r' commutes with w and its restriction ro to G is a linear projective involution commuting with wo. That is, ro commutes with

52 every 1-involution of Uq(M,h) and so must leave invariant every 1-dimensional nonisotropic subspace of G. Then by Lemma 1.2, ro(x) = x& with 8 in Z for every vector x of G. Since ro is in Uq(M,h)~ J8 = 1 and the extension r' of ro to E is a homothetic mapping of E. Thus r' is in 1, and r is in one of 1, u, v, or uv. PROPOSITION 4.7 Let v be a projective semilinear involution in rUm(Lg) relative to the automorphism a of L such that v2 = f where 3 is not of the form P = puia with M in L. If n > 4, then v(,vi) = 4. Propositions 4*1 through 4.7 yield the following theorem. THEOREM 4.1 If the dimension n of E is greater than 4 and if T is an involution of the fourth kind in PUn(Kf), then v. = 4. This means that all involutions of the fourth kind are in 3.

CHAPTER V CHARACTERIZATION OF EXTREMAL INVOLUTIONS In this chapter we characterize the extremal involutions of PUn(K,f) principally by the group-theoretical invariant at defined in Section 1.2. Thus we must determine the maximum number of elements in the set c(c(uv,,w)) where u, v and w are distinct mutually commuting involutions in ~. 5.1 Extremal Involutions We first show that a = 8 when u is an extremal involution and the dimension n of E is greater than 4. Let u be the 1-involution in u. Then every projective involution of c(u,v,w) and c(c(u,vw)) must commute with u and be of the first or third kinds by Lemma 2.1. Furthermore if they are of the first kind then they must be contained in the coset of an extremal involution by Theorems 2.1 and 2.2. Any element which commutes with u can be restricted to a transformation of U'. Since U- is nonisotropic, the form f can be restricted to a nondegenerate hermitian sesquilinear form f defined on U-. Let h be the subgroup of Un(Kf) formed of elements t which commutes with u and let, be the group of transformations t consisting of elements t of V restricted to U. Then kis contained in a Unitary Group Unl(K,f) n-l acting on U-. But every element t of Un.l(K,f) can be extended to an element t of A whose restriction is t. So 4 is the group Un_ (k,f). The sets c(u,vw) and c(c(u,v,w)) of projective involutions of Un(Kf) reduce to the sets c(v,w) and c(c(v,w)) of projective involutions of I. As elements of y2, v and w are the same kind of projective 53

involutions that v and w are. Since n-l > 3, it follows from Theorem 2.1 and Propositions 3.1, 3.3, and 3.5 that v(,w) = 4. In other words, c(c(v,w)) contains only the involutions 1, v, w and vw. Let r be an involution in c(c(uv,w). Then r in r must commute with u and hence is in 21. Therefore r can be chosen so that its restriction to U' is one of 1, v, w, or vw. Denote by r' that one of 1, v, w, or vw whose restriction to U coincides with that of r. Then r'2(x) = r2(x) = xe for all x in U' and hence for all x in E. Furthermore e is in C and e = 62 where & is in Z', the center of a subsfield K' of index 1 or 2 in K according as r is of the first or third kind and R- be the eigenspaces over K' of r and R'+ and R' be the eigenspaces over K' of r'. Since r and r' coincide on U-, we have U-r>R+ = US-R'+ and UI-R' = U-SR'-. Hence r(x) = x8 for x in R+ while r'(x) = x5 for x in R'+; r(x) = -xb for x in R and r'(x) = -x8 for x in R'-. Furthermore U+ = U+ R+ U+R-R- = U+rAR'+ IU+,R' as both r and r' commute with u. One of these summands may be zero. Indeed, if r and r' are of the first kind, then U+ = U+,R+ or U+ R' and also U1 = U+R'+ or U+R'-. Therefore for x in U+, r(x) = +x6. There are thus two possibilities for r(x) when x is in U+: r(x) = r'(x) or r(x) = -r'(x). If r and r' are of the third kind, then U+ = U+_R+ 0 U eRR = U R'+ U+,R'-. Here K' is a sfield K1 of index 2 in K and dimK1 USR* = dimK1 U+T R' = 1. Since r is in c(c(u,v,w)) while r' is in c(u,v,w), r must commute or anticommute with r'. But since they both coincide on U', r commutes with r'. Therefore we have that U is equal to U or U and U is equal to that U+ is equal to U or U>R' and U1-R- is equal to

55 U R'- or U+R'+J respectively. That is, for x in U+R+ or U+RR, r(x) = r' (x) or r(x) = -r' (x). Since U+ = U+/R+ )U++rR-, this also holds for U. Consequently the four involutions 1, v, w and vw are restrictions of at most eight involutions of c(c(6uJ,w)). Since eight is the minimal possible order of c(c(u,v,w)), we have w(u,v,W) = 8 in all cases. THEOREM 5.1 If the dimension n of E is greater than 4 and if u is an extremal involution of PUn(K,f), then cD = 8. 5.2 Involutions of the Second Kind Now let u be an involution of the second kind in PUn(K,f). Then n is even; assume n > 6. If v is another involution of the second kind, then v is conjugate to u. This is shown by Dieudonne in [4], p.7 for symplectic groups, but the same proof holds for unitary groups. Because two -elements of PUn(K,f) commute if and only if their transforms by an element t in PUn(K,f) commute, we can say that c(t rF) = t c(J )t where t1 Jt is the set of transforms by t of elements of a set J in PU (K,f). From this it follows that v- = vn for all involutions u and v of the second kind. So if one such involution is in ~, they all are.8 Thus if u is not in, we have already distinguished group-theoretically involutions of the second kind from extremal involutions. So consider that u and hence all involutions of the second kind are in &_. 8Actually it can be shown that all involutions of the second kind are in -, but,as we see here, it is not necessary to do this. I am indebted to Dieudonne for pointing this out and consequently shortening the proof.

56 Let u be in u. Then u2 = in Cn with y = k2 where X is in Z and X J = -1. Associated with u are its positive and negative eigenspaces, U+ and U., which are totally isotropic. As Dieudonn4 shows ([6], p.7), we can pick a basis xl, x2,..., xm for U+ and xm+, Xm+,..., xn for U" such that f(xixj) = 0 except when fj-il = min which case f(xixj) = 1. Then the planes Qi = xiK(xxm+iK i = 1,2,..., m, are all nonisotropic and orthogonal (cf. Section 1.4). As n must be even and n > 6, m = 4. Let P1 = xlK, P2 = x2K G X3K, P3 = x4K ~... + XmK P4 = Xm+iK, P5 = Xm+2K + Xm+3K, and Pe = Xm+4K.. ) xnK. Let V+ = P1 @ P2 ~Ps V V. = P3 ~ P4 P5, W+ = P1 @P3 P5,s and W- = P2 (P4 Pe6. Since P6 is orthogonal to P1 and P2 and P1, P2, and P6 are totally isotropic, V must be totally isotropic. Similarly V', W and W' are totally isotropic. Let v be the projective involution such that v(x) = xX for x in V+ and v(x) = -xX for x in V'. Let w be the projective involution such that w(x) = xX for x in W+ and w(x) = -xA for x in W. Then v and w are of the second kind and hence in -. Since u, v and w leave all the subspaces Pi invariant, they leave U+, U', V+, V", W+, and W- invariant; hence they all commute. We shall show that (uvi) > 8. Let rl be the 2-involution in Un(K,f) with subspaces R+ = Q1 = P1 GP4 and R1 = Q2 @ Qs ~.. Q. Let r2 be the 4-involution with subspaces R2+ = Q2 0 Qs = P2 P5 and R2 = Ql1 Q4. * * * $ Q - By choice of the vectors xi, it follows that both R1+ and R1' and both R2+ and R'2 are pairs of orthogonal nonisotropic subspaces. Since all the projective involutions u, v, w, uv, vw, wu, and uvw have eigenspaces which are each the direct sum of three of the Pi, r1 and

57 r2 are not in any of their respective cosets. So if we can show that R1 or r2 is in c(c(u,v,w)), then it follows that uo > 8. First consider that n > 10. We show that rl is in c(c(u,v,w). Then dim P3 > 2 and dim P6 > 2. Let t be a projective involution in c(u,v,w). If t commutes with u, v, and w, it leaves each Pi invariant and consequently R+ and R1 invariant; so t commutes with rl. If t anticommutes with some but not all of u, v and w, then t must map P1 onto P2, P3, P5, or P6. But this is impossible for dim P1 = 1 while Pi > 1 for i = 2, 3, 5, or 6. So the only remaining case is that t anticommutes with all of u, v,and w. Then t(P1) = P4, t(P2) = P5 and t(P3) = P6. Hence t(R1+) = R1+ and t(Rl') = R1' and t commutes with rl. Therefore rl is in c(c(u,vw)) and CD > 8. Now consider that n = 8. Then dim P3 = dim P6 = 1. Again any projective involution commuting with u, v, and w must leave R2+ and R25 invariant and consequently commutes with r2. If a projective involution commutes with some but not all of u, v, and w, then t(P2) =PI where i =, 3, 4, or 6. But dim Pi = 1 for i = l, 3, 4, or 6 while dim P2 = 2. Hence t cannot be in c(u,v,w). If t anticommutes with u, v, and w, then, as before, t(R2+) = R2+ and t(R2') = R2-. Consequently, t commutes with r2 and r2 is in c(c(u,v,w)). This again means that at > 8. We have thus THEOREM 5.2 If the dimension n of E is even and greater than 6 and if u is an involution of the second kind in PU (K,f), then cia > 8.

58 5.3 Involutions of the Third Kind Let now u be an involution of the third kind in PUn(K,f) and assume that n > 4 and n j 6. We shall show that at > 8. Let u be a representative of u. The u2 = 7 with 7 in C and 7 = X2 where X is not in Z. As in Section 1.7, let K1 be the fixed subsfield of K by k. Then K = K1 9 K1i where @ is an element of K which anticommutes with X. Then E can be considered as a K1-space E*, and E = U U where U+ and U' are the positive and negative eigenspaces of u. Set E1 = U+ and form the group rUn(Kl,fl) acting on El1p If j is a set of involutions in PUn(Kl,fl), let c*(4) denote those involutions of PUn(Ki,fl) which conmute with every involution of J. If v1 and wl are distinct commuting involutions in PUn(Ki,fl), denote the order of c*(c+(vl,wl)) by v*(viwl). In the proof of Theorem 2.2 in Section 2.3, we showed that there exists two involutions of the first kind, which we here denote by V1 and wl, such that v*(vLwl) > 4. In other words, there is an involution rl of the first kind (which is one of the ri in Section 2.5) in c*(c*(vl7,w )) which is not one of 1, vl, wl or vlwl. The restriction of u to E1 is a homothetic mapping hx in the center Cn of Un(Kifi). Let v1', w1' and rl' be the p-involutions ('p< [n/2]) contained in vl, wl, and rl, respectively; and set vi = hkvl' and wl = h.w1l' We still. have vl in vl and wl in wl; so rl is in c*(c*(vl,wl)). We can extend rl to an involution r of Un(K,f) by setting r(x@) = r1(x)g for x in E1 = U+. This defines r(x) on U = U+9 and consequently on E as E* = U+ TU'. Similarly, we can extend vl and wl

59 to projective involutions v and w in Un(K,f). But now v2 = w2 = 7; so v and w are projective involutions of the third kind in Un(Kf). Therefore, by Theorem 3.3, v and w are in. Since h\, vl and wl all commute, their extensions u, v, and w must all commute. So form c(c(u,v,w)). We shall show that r is in c(c(u,v,w)). It will then follow that v(u,v,w) > 8 as the restriction rl of r to E1 does not coincide with the restriction of any of the elements of I, u, v, w, uv, vw, wu, or uvw, which are in the cosets 1, vl, wl, and vlwl. If t is a projective involution in c(u,v,w) which commutes with u, then the restriction ti of t to El will be a projective involution in c(vl,wl). By Section 2o3, t1ir = rltl. Let x be in U+ so that x@ is in U-. Then t(r(xQ)) = tl(rl(x))Q = rl(tl(x)G = r(t(xQ)). Since E = U+ U-, t commutes with rl. If t is a projective involution in c(u,v,w) which anticommutes with u, then set t' = hat. Thus t' will commute with u and commute or anticommute with v and w. The restriction tl of t' to E1 will then commute or anticommute with vl and wl. But the subspaces Pi were chosen in Section 2.3 so that no transformation could anticommute with vl or w1. So tl commutes with vl and wl and hence leaves the eigenspaces of ri invariant. Thus tlrI = rltl. Again let x be in U+ so that xG is in Ui. Then t'(r(xg)) = -t(rl(x)Q= -rl(tl(x)Q = r(t'(xQ)). This means that t' commutes with r and so t commutes with r. Thus r is in c(c(uv,w)) and co(u,vw) > 8. THEOREM 5.3 If the dimension of E is greater than 4 and not 6 and if u is an involution of the third kind in PUn(K,f), - > 8. u

60 5.4 Involutions of the Fourth Kind Let u be an involution of the fourth kind in PUn(K,f). As mentioned in Section 1.8, n must be even. Furthermore assume n > 6. If we also assume that n / 8 and n / 12, we can conclude that at > 8 in this case by methods similar to those employed in the previous sections of this chaptero However, we can obtain the more general result having n > 6 by the following argument. Let 9 be the set of involutions u of 9 such that wD = 8. By Theorems 5o1, 5.2,and 5.3, we have that 9 contains only extremal and possibly involutions of the fourth kind. Let Wl(u) denote a maximal set of involutions of commuting with u. By Lemma 2.1, if u is extremal, )(u) contains only extremal involutions; and if u is of the fourth kind,'1 1(u) contains only involutions of the fourth kind, Let K(u) denote the order of T(u) and K; the maximum of K(u). We shall use the group-theoretical invariant sK in addition to CD- to distinguish involutions of the fourth kind from extremal involutions. Let u be an extremal involution and u a 1-involution in u. The representatives of involutions commuting with u commute with u by Lemma 2.1. Consequently, in order to determine K-, it suffices to determine the maximal number of distinct 1-involutions commuting with u. This has been shown to be n by Dieudonne ([5], p.5). Therefore, re =n for extremal involutions. Furthermore, we note that the product of any distinct two or three mutually commuting 1-involutions is a 2- or 3-involution, respectively, and is thus not in 9 by Theorem 2.2. Now in order to distinguish involutions of the fourth kind from extremal involutions, we need only consider the case that u is of the fourth kind in C with K- = n. Let ul = u, u2,..., un be the elements of a set (u) and pick representatives ui of u.. We then

61 have uiuj = +ujui. If ui2 = 7i in Cn where 7i is not a square in K, (uiuj)2 = yij = +iYj. First consider that there exists a pair i and j (i / j) such that yij is not a square in Z. Then uiuj is a projective involution of the third of fourth kind and hence in; by theorems 3.1 and 4.1i This distinguishes u from extremal involutions in this case. Otherwise 7ij = kij2 with kij in Z for all i and j where i f j. In particular, if there exists a projective involution uk within k = 3,4..., n, such that uk commutes with ulu2, then ulu2uk is of the third or fourth kind, because (ulUluk)2 = 1i22/k cannot be a square in Z without 7k being a square in Z. So in this case, ulu2uk is in. Finally suppose that all uk(k = 3,4,...,n) anticommute with ulu2. Then, as n > 6, there exists at least two, uk say u3 and u4, which commute with one, say ul,, and anticommute with the other, say u2. But then ul commutes with U3U4 and so by the above argument u1u3u4 is in. This yields the following theorem. THEOREM 5.4 If the dimension n of E is even and greater than 6, and if u is an involution of the fourth kind in PUn(K,f), u can be distinguished group-theoretically from extremal involutions by one of the three conditions: (i) u > 8 (ii) xj = 8 and iK- n (iii) w- = 8, K = n and there exists two or three involutions in a maximal commuting set yn(u) of O such that their product is in A.

62 5.5 Principal Theorem Theorems 5.1 through 5.4 imply that extremal involutions can be characterized group-theoretically in PUn(K,f)o We summarize this result in the following theorem. PRINCIPAL THEOREM 5.5 Let E be a right vector space of dimension n > 4 and n i 6 over a sfield K which possesses an involutive anti-automorphism J, which is not of characteristic 2, and which is not the finite field GF(3)o Let f be a nondegenerate hermitian sesquilinear form defined on E with respect to J and let PU (K,f) be the projective unitary group with respect to this form. Then an extremal involution u of PUn(K,f) is characterized by the conditions (i) v- = 4, Wu- =8, and KU = n. (ii) Any product of two or three involutions of' in a maximal commuting set Tl(u) of involutions of q is not in.

APPENDIX DETERMINATION OF THE AUTOMORPHISMS This appendix is included in order to state the final results that Theorem 5.5 implies. The material here is a summary of that contained in Dieudonne [5], p. 76ff., and Rickart [10]. Details of their proofs will be omitted, Dieudonne stated his result for fields but his argument extends to sfields without alteration. He required that n be finite and n > 4 while Rickart required merely that n > 6. According to Theorem 5.5, every automorphism 0 of PUn(K,f) maps 1-involutions onto 1-involutions in a one-one manner. Thus associated with 0 is a mapping 0 of P(E) which sends those elements which are 1-dimensional nonisotropic subspaces of E onto elements which are 1-dimensional nonisotropic subspaces. The problem is to show that 0 is a collineation of P(E). In fact, we have the following theorem. Theorem A.1 (Rickart's Theorem 2.6). There exists a one-one mapping 0 of the projective space P(E) onto itself with the following properties: (i) if n is an extremal involution in PUn(K,f) with associated 1-dimensional subspace xK, then O(xK) is the 1-dimensional subspace associated with 0(G). (ii) O(xK) is orthogonal to (yK) if and only if xK is orthogonal to yK. (iii) O(zK) c O(xK) ~ 0(yK) if and only if zK C xK ~ yK. Proof If xK is a nonisotropic subspace and u is the 1-invOlution with xK as its positive eigenspace, define 6(xK) to be the positive 65

64 eigenspace of 0(u). If xK is isotropic, then it can be characterized as the intersection of two nonisotropic planes, P1 and P2. These planes are the subspaces of two 2-involutions v- and v2, Two such 2-involutions are called intersecting 2-involutions if the subspace generated by their 2dimensional eigenspaces is 3-dimensional. Not only 2-involutions but 3involutions can be characterized group-theoretically (that is, in terms of 1-involutions). Those intersecting 2-involutions for which the common part of their 2-dimensional subspaces is isotropic can be characterized grouptheoretically by having no 1-involution commuting with both of them. So we can define O(xK) to be the intersection of the 2-dimensional subspaces of 0(ev) and 0(V2). The next step is to show that 0 is single valued. Clearly < (xK) is uniquely defined if xK is nonisotropic. If xK is isotropic, then it must be shown that O(xK) is defined independently of the pairs of intersecting 2-involutions whose 2-dimensional subspaces intersect in xK, This is the crucial part of the proof and the principal place where the proofs of Dieudonne and Rickart differ. The details of these proofs will be omitted here. Property (i) is immediate. Property (ii) is trivial for nonisotropic subspaces because orthogonal subspaces belong to commuting involuutions. For isotropic subspaces, this is proved by taking appropriate linear combinations of vectors involved so that the problem is reduced to the former cases Property (iii) is proved by contradiction. Suppose that O(zK) 0 $(xK) @ O(yK) while zK c xK e yK, Then there exists a vector Z' such that O(z'K) is orthogonal to 4(xK) and 0(yK) but not to O(zK). This implies that zK is orthogonal to x and yK but not to zK by (ii), which is a contradiction,

65 Now apply the fundamental theorem of projective geometry to obtain the existence of a semilinear transformation r relative to an automorphism p of Kauchkthat $(xK) = r(x)K, We show that r is in rUn(Kf). Let x and y be arbitrary nonzero vectors in E and pick z such that f(x,z)=i. Set f(x,y) = 5 and form y' = y - z~. Then f(x,y') = 0 and so f(r(x),r(y'))=0. Hence f(r(x),r(y)) = f(r(x),r(z))-f(x,y)P. Set ex = f(r(x),r(z)). Then ex is shown to be independent of x by using the additivity of f(x,y). Thus ex = e is the multiplier of r in rUn(Kf). Let r be the image of r under the natural homomorphism of rTi(K,f) onto P PUn(K,f). Then we have Theorem A.2'When PUn(K,f) is subject to the conditions of Theorem 5.5, every automorphism 0 of PUn(K,f) is of the form 0 t -rtrr where r is an element of PrUn(K,f). Indeed, if u is an extremal involution, we have 0(u) = -ir1. If E is in PTLn(K,f), then tut- is also an extremal involution. Therefore 0({tf-1) = rtutl"1 = 0(t)ruril0(t)l. So h uh = u where h = -l 0(). On the other hand, F(fl)f- is easily verified to be in PUn(K,f); so 5 is in PUn(K,f). Let h be in h and u be the 1-involution in u. Then h is an element of Un(Kf) which commutes with every l-involution. Thus by Lemma 1.2, h is a homothetic mapping. Hence h = 1. This implies that 0(t) = rtr1. Thus we have characterized the automorphisms of PUn(Kf). As Rickart points out, the automorphism 0 can be considered just as well to be an isomorphism between two projective unitary groups PUn(Kf) and PUm(Lg) acting on two projective spaces P(E) and P(F). Then r is a semilinear transformation of E onto F relative to an isomorphism a of K onto L such that g(r(x),r(y)) = ef(xy)a for x and y in E. Finally F is

65 the induced collineation between P(E) and P(F). Furthermore, he notes that PUn(Kf) and PUm(L,g) can be replaced by subgroups which contain all the involutions of these groups since to characterize 0 in the manner of Theorem A.2 only requires knowledge of its effect on the extremal involutions of PUn(K,f), which in turn need to be- distinguished only from other involutions of PUn(K,f). This interpretation has interesting applications in the infinite dimensional case.

BIBLIOGRAPHY 1. G. Birkhoff and J. von Neumann, The Logic of Quantum Mechanics, Annals of Mathematics, vol 537 823-43 (193)o 2. J. Dieudonne, 4* - 4._ iii5.i i. ____ii., 6 L. K. Hua, 7. G. W. Mackey, La Geometrie des groupes classiques, to appear in Ergebnisse der Mathematik, Les Extension quadratiques des corps non commutatifs et luers applications, Acta Mathematica, vol 87,175-242 Sur les group classiques, Actualitees Scientifiques et Industrielles, No, 10o40, Paris, Hermann and Cie, (1950), On the Automophisms of the Classical Groups, Memoirs of the American Mathematical Society, No, 2 (1950), Supplement to the Paper of Dieudonne on the Automorphisms of the Classical Groups, Memoirs of the American Mathematical Society, No. 2 (1950). Isomorphisms of Normed Linear Spaces, Annals of Mathematics, vol 435, 244-260 (19422) 8. 0. Schreier and B. L. van der Waerden, Die Automorphismen der Projektiven Gruppen, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universitat, vol 6, 303-322 (1928). 9, C. E. Rickart, Isomorphic Groups of Linear Transformations, American Journal of Mathematics, vol 72 I 451-464(T950). 10*,Isomorphic Groups of Linear Transformations, II, American Journal of Mathematics, vol 73, 697-716 (1951), 67

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