Let A and B be two Abelian groups, and define the sum of two homomorphisms η and χ from A to B by a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ . With this addition, the set of homomorphisms from A to B forms an Abelian group H. Suppose now that A is a p-group ( p a prime number). Prove that in this case H becomes a topological group under the topology defined by taking the subgroups p^kH \;(k\equal{}1,2,...) as a neighborhood base of 0. Prove that H is complete in this topology and that every connected component of H consists of a single element. When is H compact in this topology? [L. Fuchs] abstract algebratopologygroup theorylimitgeometrygeometric transformationsuperior algebra