Isoclinism and stable cohomology of wreath products

TL;DR

This paper explores the stable cohomology of finite p-groups, especially wreath products of cyclic groups, using isoclinism, and provides explicit descriptions and applications to groups of Lie type.

Contribution

It introduces methods to determine stable cohomology via isoclinism and abelian subgroups, including explicit calculations for wreath products of cyclic groups.

Findings

Stable cohomology of many p-groups is detected by abelian subgroups.

Explicit cohomology algebra for wreath products of cyclic groups is provided.

Applications to unramified and stable cohomology of Lie type groups are discussed.

Abstract

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups, see Theorem 4.4. Moreover, we show that the stable cohomology of the n-fold wreath product of cyclic groups ZZ/p is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.