Introduction

If G is a group then as usual we denote the automorphism group of G by Aut G. In this paper we give a survey of results concerning Aut G in the case when G is nilpotent or satisfies some generalized form of nilpotency. The first part of the paper will be concerned with the situation in which Aut G is as small as it can be. For each group G we let Z(G) denote the centre of G, although when no ambiguity arises we shall often simply denote Z(G) by Z. As usual we write G′ to denote the derived subgroup of G. Of course, G/Z(G) is isomorphic to Inn G, the group of inner automorphisms of G. In Section 2, we discuss the situation in which Inn G = Aut G. We give rather an old example due to Zaleskii [50] showing that this situation can hold for nilpotent groups and then discuss some consequences.

As usual we denote the factor group Aut G/ Inn G by Out G, the group of outer automorphisms of G. Thus Section 2 is concerned with the situation when Out G = 1. By contrast, in Section 3 we discuss the situation which in a sense is opposite to that of Section 2. In Section 3 we are interested in showing that often Out G is rather large. The discussion is concerned mostly with nilpotent p-groups in Section 3. However we also discuss recent work of Puglisi and some simple consequences.