A note on abelian subgroups of maximal order

In a recent paper of the first author and I. M. Isaacs it was shown that if m = m(G) is the maximal order of an abelian subgroup of the finite group G, then |G| divides m! ([AI18, Thm. 5.2]). The purpose of this brief note is to improve on the m! bound (see Theorem 2.1 below). We shall then take up the task of determining when the (implicit) inequality of our theorem becomes an equality. Despite, perhaps, first appearances this determination is not trivial. To accomplish it we shall establish a result (Theorem 2.3) of independent interest and we shall then see that Theorems 2.1 and 2.3 combine to further strengthen Theorem 2.1 (see Theorem 3.4).