In this paper, we consider the influence that the maximal size of an abelian subgroup of a group exerts on the size of the group. We will first prove that divides , the product of all prime powers at most . We then show that if a prime divides then either 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 holds only when is a small symmetric group and to derive an explicit upper bound for as a function of . We conclude our paper by determining the order of magnitude of this upper bound.
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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. 319–334DOI 10.4171/RLM/893