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arXiv · 2206.03403

Criteria for solubility and nilpotency of finite groups with automorphisms

Cristina Acciarri·Robert M. Guralnick·Pavel Shumyatsky
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Abstract

Let $G$ be a finite group admitting a coprime automorphism $α$. Let $J_G(α)$ denote the set of all commutators $[x,α]$, where $x$ belongs to an $α$-invariant Sylow subgroup of $G$. We show that $[G,α]$ is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set $J_G(α)$ is soluble or nilpotent, respectively.

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Cristina Acciarri, Robert M. Guralnick, Pavel Shumyatsky. 2022-11-01. Criteria for solubility and nilpotency of finite groups with automorphisms. https://arxiv.org/abs/2206.03403

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