THE AUTOMORPHISM GROUP OF A FINITE MINIMAL NON-ABELIAN p-GROUP

Determining the order and the structure of the automorphism group of a finite p-group is an important problem in group theory. There have been a number of studies of the automorphism group of p-groups. Most of them deal with the order of Aut(G), the automorphism group of G, see for example [1] and [6]. Moreover various attempts have been made to find a structure for the automorphism group of a finite p-group, see [3] and [5]. In this paper we study the automorphism group of a finite minimal nonabelian p-group. A minimal non-abelian group is a non-abelian group such that all its proper subgroups are abelian. A presentation of these groups is given in ([2], §1, Exercise 8a). Let G be a finite minimal non-abelian p-group. Some results about the order of Aut(G) is given in [6] when p > 2. In this paper first we find the order of Autc(G), the central automorphism group of G, and then we give a structure theorem for Autc(G). Moreover for p = 2 we find the order of Aut(G) and we show that Aut2(G), the 2-Sylow subgroup of Aut(G), is a split extension of Autc(G) by Z2, where Z2 is the cyclic group of order 2. Finally we give a structure theorem for Aut(G) when p = 2 (see Theorem 3.7). In particular we study a problem posed by Y. Berkovich ([2], Problem 700). Throughout this paper, the following notation is used. p denotes a prime number. We use d(G) for the minimal number of generators of a group G. The