Dikran Karagueuzian

The module structure of a group action on a polynomial ring: A finiteness theorem

Consider a group acting on a polynomial ring over a finite field. We study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries. For example, any indecomposable module that appears as a summand must also appear in low degree, and so the number of isomorphism types of such summands is finite. There are also applications to invariant theory, giving a priori bounds on the degrees of the generators.

Essential cohomology of finite groups

Abstract. In this paper we prove that a finite group G with Cohen-Macaulay mod p cohomology will have non-trivial undetectable elements in

Involutions and characters of upper triangular matrix groups

H^*(G,{\Bbb F}_p)

The cohomology of the Sylow 2-subgroup of the Higman–Sims group

if and only if G is a p-group such that every element of order p in G is central. Applications and examples are also provided.

Topological models and cohomology of Galois groups

We study the realizability over R of representations of the group U(n) of upper-triangular n x n matrices over F 2 . We prove that all the representations of U(n) are realizable over R if n ≤ 12, but that if n ≥ 13, U(n) has representations not realizable over R. This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-Lopez and of the authors, but the proof of the theorem in this paper is much more natural.

Conjugacy in Groups of Upper Triangular Matrices

Abstract In this paper we compute the mod 2 cohomology of the Sylow 2-subgroup of the Higman–Sims group HS, one of the 26 sporadic simple groups. We obtain its Poincare series as well as an explicit description of it as a ring with 17 generators and 79 relations.

On the Cohomology of Galois Groups Determined by Witt Rings

Abstract Let F denote a field of characteristic different from 2. In this Note we describe the mod 2 cohomology of a Galois group ς F which is determined by the Witt ring W F .