Derived Subgroups of Fixed Points in Profinite Groups

TL;DR

This paper proves that if a prime q and an elementary abelian group A of order q^3 act on a profinite group G with certain periodicity conditions on centralizers, then the derived subgroup G' is locally finite.

Contribution

It establishes a new criterion linking the periodicity of centralizers under automorphisms to the local finiteness of the derived subgroup in profinite groups.

Findings

G' is locally finite under the given conditions.

Periodic centralizers imply local finiteness of the derived subgroup.

Provides new insights into automorphism actions on profinite groups.

Abstract

The main result of this paper is the following theorem. Let q be a prime, A an elementary abelian group of order q^3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that C_G(a)' is periodic for each nontrivial element a in A. Then G' is locally finite.