Exponent of a finite group admitting a coprime automorphism

2019/07/04 by Rodrigues, Sara, Shumyatsky, Pavel
#20D45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1907.02396

Abstract

Let G be a finite group admitting a coprime automorphism ϕ of order n. Denote by G_ϕ the centralizer of ϕ in G and by G_-ϕ the set \ x^-1x^ϕ; x∈ G\. We prove the following results. 1. If every element from G_ϕ∪ G_-ϕ is contained in a ϕ-invariant subgroup of exponent dividing e, then the exponent of G is (e,n)-bounded. 2. Suppose that G_ϕ is nilpotent of class c. If x^e=1 for each x ∈ G_-ϕ and any two elements of G_-ϕ are contained in a ϕ-invariant soluble subgroup of derived length d, then the exponent of [G,ϕ] is bounded in terms of c,d,e,n.

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