MathDB

Let

A

and

B

be two Abelian groups, and define the sum of two homomorphisms

\eta

and

\chi

from

A

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

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]

Problem Statement