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Generalized torsion elements in groups

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Abstract

A group element is called a generalized torsion element if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the generalized order of an element in a finite group G using its character table.

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Acknowledgements

We are very grateful to the anonymous referee for the many suggestions which improved the first version of the article. In particular, the current definition of the generalized exponent and the statement of Proposition 2.2(5) were suggested by him or her. We are also grateful to T. Ito for useful discussions regarding this paper. This work was partially supported by CNPq and FAPDF (Brazil). The second author acknowledges the financial support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.: 421624/2018-3 and 402934/2021-0) and the FAPEMIG Project Universal (project no.: APQ-00971-22). The third author was partially supported by FAPEMIG RED-00133-21.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, Campus Universitário Darcy Ribeiro, 70910-900, Brasilia, DF, Brazil

    Raimundo Bastos

  2. Departamento de Matemática, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, MG, Brazil

    Csaba Schneider

  3. Departamento de Ciências Exatas e Aplicadas, Universidade Federal de Ouro Preto, 35931-008, Ouro Preto, MG, Brazil

    Danilo Silveira

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  1. Raimundo Bastos
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  2. Csaba Schneider
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  3. Danilo Silveira
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Correspondence to Danilo Silveira.

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Bastos, R., Schneider, C. & Silveira, D. Generalized torsion elements in groups. Arch. Math. 122, 121–131 (2024). https://doi.org/10.1007/s00013-023-01931-5

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