Rank, Coclass, and Cohomology

Abstract

We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of fixed rank.

Cohomology of profinite groups of bounded rank

“…Now we prove Theorem 1.1. This is proved for finite ‐groups in [6, 1.1] and for pro‐ groups in [6, 1.4]. Let be a finite group with Sylow ‐subgroup .…”

“…First we prove part (1) of Theorem 1.2. This is proved for finite ‐groups in [6, 1.2]. If is a finite group with Sylow ‐subgroup , then a standard transfer argument shows that the restriction map embeds in , so the bound hold for .…”

“…The dimension of the sum of the cohomology groups in degrees 0 through is bounded in terms of and , by Theorem 1.2(1), and this bounds the number of generators needed. The proof of [6, 1.3] now applies verbatim to prove part (2) of 1.2.…”

“…The purpose of this note is to generalise to profinite groups our results of [6] for pro‐ groups of bounded rank. The main one of these states that amongst the pro‐ groups of rank bounded by a number there are only finitely many mod‐ cohomology rings up to isomorphism.…”

Cohomology of profinite groups of bounded rank

“…The purpose of this note is to generalise to profinite groups the results of [5] for pro-p groups of bounded rank. The principal one of these states that amongst the pro-p groups of rank bounded by a number r there are only finitely many mod-p cohomology rings up to isomorphism.…”