Nilpotents of representation rings of finite p-groups

by 1968- Hindman, Peter Blake

The representation ring a(G) of a finite group G provides a context in which to study the behavior of the module category under tensor product. Much work has been devoted to the semisimplicity question for representation rings, by studying the existence and degree of nilpotents in a(G). In this paper, we construct a nilpotent of degree 3 in the representation ring a(Z/3 × Z/3), apparently the first explicit construction of such an element in odd characteristic. We make a number of observations on the general nilpotence question, and discuss applications of techniques developed in this paper to related questions. Index words: Modular Representation Theory, Relative Cohomology, Group Cohomology Nilpotents of Representation Rings of Finite p-Groups by Peter Blake Hindman A.B., The University of Chicago, 1989 M.S., Florida Atlantic University, 1993 A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2002 c? 2002 Peter Blake Hindman All Rights Reserved Nilpotents of Representation Rings of Finite p-Groups by Peter Blake Hindman Approved: Major Professor: David J. Benson Committee: Sybilla K. Beckmann-Kazez Jon F. Carlson Leonard Chastkofsky Kenneth Johnson Electronic Version Approved: Gordhan L. Patel Dean of the Graduate School The University of Georgia August 2002