Abelian Groups pp 213-228 | Cite as

# Homomorphism Groups

## Abstract

The fact that the homomorphisms of a group into another group form an abelian group has proved extraordinarily profound not only in abelian group theory, but also in Homological Algebra where the functor Hom is one of the cornerstones of the theory. Our first aim is to find relevant properties of Hom both as a bifunctor and as a group.

It is rather surprising that in some significant cases \(\mathop{\mathrm{Hom}}\nolimits (A,C)\) is algebraically compact; for instance, when *A* is a torsion group, or when *C* is algebraically compact. In the special situation when *C* is the additive group \(\mathbb{T}\) of the reals mod 1, in which case \(\mathop{\mathrm{Hom}}\nolimits (A, \mathbb{T})\), furnished with the compact-open topology, will be the character group of *A*, our description leads to a complete characterization of compact abelian groups by cardinal invariants. An analogous result deals with the linearly compact abelian groups.

The final section discusses special types of homomorphisms that play important roles in the theory of torsion groups.