## 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.## References

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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