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Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

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Abstract

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O p (G) = 1 then |G/F(G)| p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and k p-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).

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Authors and Affiliations

  1. Departament d’Àlgebra, Universitat de València, 46100, Burjassot, València, Spain

    Alexander Moretó

  2. Department of Mathematics, The University of Akron, Akron, Ohio, 44325, USA

    Hung Ngoc Nguyen

Authors
  1. Alexander Moretó
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  2. Hung Ngoc Nguyen
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Corresponding authors

Correspondence to Alexander Moretó or Hung Ngoc Nguyen.

Additional information

The research of the first author was supported by the Spanish Ministerio de Ciencia y Tecnología, grant MTM2010-15296 and PROMETEO/Generalitat Valenciana.

The second author is partially supported by the NSA Young Investigator Grant #H98230-14-1-0293 and a BCAS Faculty Scholarship Award from the Buchtel College of Arts and Sciences, The University of Akron.

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Moretó, A., Nguyen, H.N. Variations of Landau’s theorem for p-regular and p-singular conjugacy classes. Isr. J. Math. 212, 961–987 (2016). https://doi.org/10.1007/s11856-016-1316-7

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  • DOI: https://doi.org/10.1007/s11856-016-1316-7

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