Groups of exponent 4 as automorphism groups

Abstract

If a finite group A of exponent 4 is the automorphism group of a torsionfree (abelian) group G, then A must be a subdirect product of cyclic groups C2 and C4 and of quaternion groups Qs (see [2, 4]); A is therefore contained in the variety V(Qs) generated by the quaternion group. Having dealt completely in [3] with the case of finite (or countable) abelian automorphism groups, we now turn to non-abelian groups (which, as we have mentioned before ([3], p. 34), must be of exponent 4 or 12). In this paper we confine ourselves to proving that for every n>2 the free n-generator group A, of V(Qs) is, in fact, the automorphism group of a suitably chosen torsionfree group 1. We also give at the end of the paper an example (with n = 3) of an epimorphic image of Aa of order 27 that cannot be an automorphism group, although it still satisfies the condition N3 of [2]: all its elements of order 2 lie in its centre. This group fails to satisfy the necessary condition N5 of [1] and cannot be a subdirect product of the kind described above.