Elementary abelianp-groups as automorphism groups of infinite groups. I

Abstract

If G is a group we will write Aut G for the group of all automorphisms of G and Inn G for the normal subgroup of all inner automorphisms of G. Many authors have studied the relationship between the structure of G and that of Aut G, in particular when the latter is finite. This paper is a further contribution to this study. The first results on groups whose automorphism groups are finite were published by Baer in a paper [2] in which he proved that a torsion group has finite automorphism group only if it is finite. Baer also proved that a group with only a finite number of endomorphisms is finite. In 1962 Alperin [1] characterized finitely generated groups with finitely many automorphisms as finite central extensions of cyclic groups. Nagrebeckil [9] discovered in 1972 the important result that in any group with finitely many automorphisms the elements of finite order form a finite subgroup. This of course generalizes Baer's original result. Robinson El0] has given another proof of Nagrebeckfi's Theorem as well as obtaining information on the primes dividing the order of the maximal torsion subgroup. He also characterized the center of a group whose automorphism group is finite and gave a general method for constructing examples. On the other hand there seems to be little hope of obtaining a useful classification of groups whose automorphism groups are finite, even in the abelian case. Indeed, it has been shown be several authors that torsion-free abelian groups with only one non-trivial automorphism the involution x F--~x~ are relatively common (de Groot [5], Fuchs [4], Corner [3]). However, Hallett and Hirsch have adopted a different approach, asking which finite groups can occur as the automorphism groups of torsion-free abelian groups. They have established the following definitive result [-7, 8]: