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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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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Additional Information
  • Cristina Acciarri
  • Affiliation: Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil
  • Address at time of publication: Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/b I-41125 Modena, Italy
  • MR Author ID: 933258
  • ORCID: 0000-0002-7895-7705
  • Email: cristina.acciarri@unimore.it
  • Robert M. Guralnick
  • Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
  • MR Author ID: 78455
  • ORCID: 0000-0002-9094-857X
  • Email: guralnic@usc.edu
  • Pavel Shumyatsky
  • Affiliation: Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil
  • MR Author ID: 250501
  • ORCID: 0000-0002-4976-5675
  • Email: pavel@unb.br
  • Received by editor(s): August 3, 2020
  • Received by editor(s) in revised form: July 29, 2021
  • Published electronically: April 21, 2022
  • Additional Notes: The first and the third authors were supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF), Brazil. The second author was partially supported by a Simons Foundation fellowship and NSF grant DMS-1901595
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4549-4565
  • MSC (2020): Primary 20D45
  • DOI: https://doi.org/10.1090/tran/8553
  • MathSciNet review: 4439485