Torsion-free groups having finite automorphism groups. I

Abstract

If G is an infinite periodic group, then its automorphism group is also infinite (Baer [I]); if G, in addition, is abelian, then more detailed information is available on the cardinal number of Aut (G) (Boyer [2]; Walker [13]). But in contrast, if G is torsion-free, then Aut (G) may well be a finite group. The simplest example shows this: the infinite cyclic group C, , which has only one automorphism other than the identity. The problem we shall discuss in this paper is the following: for what finite groups A is there a torsion-free group G such that Aut (G) is isomorphic to A ?’ We remark immediately that under these circumstances G is necessarily abelian. For if Aut (G) is finite, then so is its subgroup consisting of the inner automorphisms, which is isomorphic to the factor group of G over its center Z(G). But by a celebrated theorem of Schur,2 if the center of a group is of finite index, then its derived group G’ is finite. And in our case, since G is torsion-free, this means that G’ : 1 or that G is abelian. We do not concern ourselves with the apparently hopeless task of finding all the torsion-free abelian groups whose automorphism group is a given finite group. It may suffice here to state that if a finite group A occurs at all, then it will become clear from the examples we shall construct in Part II that even among countable torsion-free abelian groups G of finite rank there are always uncountably many nonisomorphic ones having the given A as their automorphism group. In fact, much more is known even in the simplest case when A F C, is cyclic of order 2. Preliminary results by de Groot [7], Hulanicki [IO], Fuchs [5], and Saqiada [II] showed successively that for every cardinal