Large Abelian subgroups of 𝑝-groups

Abstract

For some time now, very little has been known about abelian subgroups of pgroups.We shall try and remedy this situation in a series of papers, of which this is the first. One impetus for doing this was created by several natural conjectures which arose in problems in other areas of group theory. In this paper we shall study p-groups with respect to the existence and nonexistence of abelian subgroups which in one sense or another can be considered as large subgroups. Later work will deal with several other topics. Our first result is in a negative direction; we shall demonstrate the falsity of the conjecture that every group of order pn has an abelian subgroup of order pn/2. Previously, we announced [1] that the best one could hope for was abelian subgroups of order at least p*48n, but for odd primes we can greatly improve this result by an entirely different method.