A note on abelian subgroups of maximal order

Abstract

In this paper, we consider the influence that the maximal size $m$ of an abelian subgroup of a group exerts on the size of the group. We will first prove that $|G|$ divides $g(m)$, the product of all prime powers at most $m$. We then show that if a prime $p>m/2$ divides $|G|$ then either $G$ 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 $|G|=g(m)$ holds only when $G$ is a small symmetric group and to derive an explicit upper bound for $|G|$ as a function of $m$. We conclude our paper by determining the order of magnitude of this upper bound.