On the Automorphism Group of a Finite p-Group with a Small Central Quotient

Abstract

In recent years there has been considerable interest in the conjecture that |G| divides |Aut G| for all finite non-cyclic p-groups G of order greater than p2. In particular, the conjecture has been established for a considerable number of (not necessarily distinct) classes of finite p-groups ([6], [7], [8], [9], [15], [16]); additionally, results have been obtained, often using homological methods, which permit reductions in any attempt to establish the overall conjecture ([5], [10], [13], [15]). In the former case, the p-groups G have generally been regular p-groups (see, for example, [6]) and the prime p = 2 has either been excluded (see, for example, [8]) or treated as a special case (as in [9]).It is the purpose of this paper to establish the conjecture for the class of all p-groups G where |G: Z(G)| ≦ p4 with no restrictions on the prime p.