Coprime automorphisms of finite groups

Acciarri, Cristina; Guralnick, Robert; Shumyatsky, Pavel

doi:10.1090/tran/8553

Coprime automorphisms of finite groups
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by Cristina Acciarri, Robert M. Guralnick and Pavel Shumyatsky PDF

Trans. Amer. Math. Soc. 375 (2022), 4549-4565 Request permission

Abstract:

Let $G$ be a finite group admitting a coprime automorphism $\alpha$ of order $e$. Denote by $I_G(\alpha )$ the set of commutators $g^{-1}g^\alpha$, where $g\in G$, and by $[G,\alpha ]$ the subgroup generated by $I_G(\alpha )$. We study the impact of $I_G(\alpha )$ on the structure of $[G,\alpha ]$. Suppose that each subgroup generated by a subset of $I_G(\alpha )$ can be generated by at most $r$ elements. We show that the rank of $[G,\alpha ]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(\alpha )$ has odd order, then $[G,\alpha ]$ has odd order too. Further, if every pair of elements from $I_G(\alpha )$ generates a soluble, or nilpotent, subgroup, then $[G,\alpha ]$ is soluble, or respectively nilpotent.

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