Coprime automorphisms of finite groups

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 su...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2022-07, Vol.375 (7), p.4549-4565
Main Authors: Acciarri, Cristina, Guralnick, Robert, Shumyatsky, Pavel
Format: Article
Language:English
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8553