Abelian Groups pp 655-671 | Cite as

# Automorphism Groups

## Abstract

Needless to say, automorphism groups contain in general much less information about the group structure than the endomorphism rings. There is an enormous contrast between torsion and torsion-free groups, perhaps even larger than from the point of view of endomorphism rings. Here again, the case of *p*-groups is more favorable (at least for *p* > 2) inasmuch as their automorphism groups determine the groups up to isomorphism. The torsion-free case seems uncontrollable, but a close examination shows that, interestingly, only a handful of finite groups may occur as automorphism groups of torsion-free groups of finite rank.

We remark at the outset that we will not give complete proofs of the two most important theorems on automorphism groups: that \(\mathop{\mathrm{Aut}}\nolimits A\) determines *A* if *A* is a *p*-group (*p* > 2), and the full description of finite automorphism groups of finite rank torsion-free groups. The proofs are too long and require lot of arguments on non-commutative groups.