Fixed points of coprime operator groups

TL;DR

This paper investigates the fixed points of coprime automorphism groups acting on finite groups, establishing bounds on the exponents of certain derived and lower central series subgroups under specific conditions.

Contribution

It provides new bounds on the exponents of derived and central series subgroups of finite groups acted upon by coprime elementary abelian groups, extending previous results in group theory.

Findings

Bounded exponent for the dth derived subgroup of the centralizer

Bounded exponent for the (r-1)th lower central subgroup

Results depend on the action of elementary abelian groups

Abstract

Let m be a positive integer and A an elementary abelian group of order q^r with r greater than or equal to 2 acting on a finite q'-group G. We show that if for some integer d such that 2^{d} is less than or equal to (r-1) the dth derived group of C_{G}(a) has exponent dividing m for any nontrivial element a in A, then $G^{(d)}$ has {m,q,r}-bounded exponent and if $\gamma_{r-1}(C_G(a))$ has exponent dividing m for any nontrivial element a in A, then $\gamma_{r-1}(G)$ has {m,q,r}-bounded exponent.