A note on abelian subgroups of maximal order

  • Stefanos Aivazidis

    Stockholm, Sweden
  • Robert M. Guralnick

    University of Southern California, Los Angeles, USA
A note on abelian subgroups of maximal order cover

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In this paper, we consider the influence that the maximal size mm of an abelian subgroup of a group exerts on the size of the group. We will first prove that G|G| divides g(m)g(m), the product of all prime powers at most mm. We then show that if a prime p>m/2p > m/2 divides G|G| then either GG is almost simple or of very restricted type and we determine the complete list of finite simple groups with exactly one such "large" prime divisor. We are then able to deduce that G=g(m)|G|=g(m) holds only when GG is a small symmetric group and to derive an explicit upper bound for G|G| as a function of mm. We conclude our paper by determining the order of magnitude of this upper bound.

Cite this article

Stefanos Aivazidis, Robert M. Guralnick, A note on abelian subgroups of maximal order. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 2, pp. 319–334

DOI 10.4171/RLM/893