ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR QUARTERLY REPORT NO, 1 (Covering Period November 1, 1953 - January 31, 1954) By ROBERT M. THRALL Project Director JAMES P. JANS JOHN 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 March, 1954

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN QUARTERLY REPORT NO. 1 INTRODUCTION Work on this project began on a small scale in November and with somewhat enlarged scope during December and January. Individual reports of the two current project workers follow. Plans are completed for the fullscale work proposed for the summer period. Both John Walter and James Jans expect to complete their dissertations during the spring semester. PROGRESS REPORT OF JAMES P. JANS During the psst three months the inequivalent indecomposible representations of algebras over an algebraically closed field have been investigated. A sufficient condition on the lattice of two-sided ideals in the radical for there to exist an infinite number of degrees such that the algebra has infinitely many inequivalent representations of each such degree. If the algebra is commutative this condition is also necessary, for if the condition fails in a commutative algebra, the algebra has only a finite number of inequivalent indecomposible representation. PROGRESS REPORT OF JOHN WALTER Work for the past three months has centered on two activities. First, the author's thesis on the automorphism of the projective unitary groups is being completed. The automorphism of these groups acting on linear spaces of dimension greater than 6 and over skew fields which are not of characteristic 2 and have more than three elements have been determined. J - - 1

7- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - This extends the work of Dieudonne and Rickart and also provides a more systematic approach to the discussion of the automorphisms of other Classical Groups. If is expected that 4 to 6 weeks longer will be required to prepare the final draft. Second, papers by Thrall, Neabitt, and Brauer on representation theory have been read and at the suggestion of Professor' Thrall a paper by K. Taketa on metabelian groups which contains information about commutative group algebras over a finite field has been investigated. Future work will be directed toward determining the structure of commutative algebras. Taketa's article will serve as a good starting point, since he determined canonical forms for the indecomposable components of a representation of commutative group algebras over a finite field. An attempt will be made to extend his results and relate these results to intrinsic properties of algebras. 2

DISTRIBUTION LIST 3 Office of Ordnance Research Box CM, Duke Station Durham, North Carolina 2 Office, Chief of Ordnance Washington 25, D. C. Attention: ORDTB-PS 2 Chief, Detroit Ordnance District 574 East Woodbridge Detroit 51, Michigan Attention: ORDEF-IM 3