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On products of groups and indices not divisible by a given prime

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Abstract

Let the group \(G = AB\) be the product of subgroups A and B, and let p be a prime. We prove that p does not divide the conjugacy class size (index) of each p-regular element of prime power order \(x\in A\cup B\) if and only if G is p-decomposable, i.e. \(G=O_p(G) \times O_{p'}(G)\).

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Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada (IUMPA-UPV), Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia, Spain

    María José Felipe, Ana Martínez-Pastor & Víctor Sotomayor

  2. Department of Mathematics, Yaroslavl P. Demidov State University, Sovetskaya Str 14, Yaroslavl, Russia, 150014

    Lev S. Kazarin

Authors
  1. María José Felipe
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  2. Lev S. Kazarin
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  3. Ana Martínez-Pastor
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  4. Víctor Sotomayor
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Corresponding author

Correspondence to Ana Martínez-Pastor.

Additional information

Communicated by John S. Wilson.

Dedicated to the memory of Carlo Casolo

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Research supported by Proyecto PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovación y Universidades, Spain, and FEDER. The second author is also supported by Project VIP-008 of Yaroslavl P. Demidov State University and the third author by Proyecto PROMETEO/2017/057 from the Generalitat Valenciana, Spain.

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Felipe, M.J., Kazarin, L.S., Martínez-Pastor, A. et al. On products of groups and indices not divisible by a given prime. Monatsh Math 193, 811–827 (2020). https://doi.org/10.1007/s00605-020-01446-z

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  • DOI: https://doi.org/10.1007/s00605-020-01446-z

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