Central automorphisms of periodic groups

1. Introduction. Ifp is a prime and G is an additive abelian p-group, then G is a p-adic module and multiplication by a p-adic unit induces an automorphism in the center Z(Aut G) of the full automorphism group of G. It was shown by Baer ([2]; see also Theorem 115.1 of [6]) that, except possibly when p = 2, all elements of Z(Aut G) are induced this way. (The exceptional cases when p = 2 were determined precisely by Baer and in these, the group of such automorphisms is a direct factor of index two in Z (Aut G).) This note represents an attempt to extend Baer's theorem to the situation in which G is non-abelian. In multiplicative terminology (which will be used throughout this paper), automorphisms of the type described above are examples of power automorphisms - that is, automorphisms which leave every subgroup invariant. (See, for example, [4] .) The main results of this paper (Theorems A and B) will show that if G is a periodic group and Ac(G ) is the group of central automorphisms of G (that is, those which commute with all inner automorphisms), then G may be factored into two sections, one on which the whole of Ac(G) acts trivially and another which is central and on which Z(Ac(G)) induces a group of power automorphisms. There are no exceptional 2-groups mentioned in the statements of these results (although 2-groups do apparently require somewhat more delicate treatment in the actual proof) and so it is clear that they can be at best partial generalizations of Baer's theorem. On the other hand, if G is an abelian p-group with p odd, inversion is an automorphism with no non-trivial fixed points and hence, Baer's theorem is, in this situation, a special case of our results. In the final section, Theorems A and B are applied to obtain some insight into the possible structure of the automorphism group of a periodic group.