TITLE: Example of profinite groups
QUESTION [5 upvotes]: Could someone help me with an simple example of a profinite group that is not the p-adics integers or a finite group? It's my first course on groups and the examples that I've found of profinite groups are very complex and to understand them requires advanced theory on groups, rings, field and Galois Theory. Know a simple example?
Last, how to prove that that $\mathbb{Z}$ not is a profinite group?

REPLY [3 votes]: One way to get a profinite group is to start with any torsion abelian group $A$, and take $\hom(A, \mathbb{Q}/\mathbb{Z})$. This acquires a topology as the inverse limit topology: it is the inverse limit of $\hom(A_0, \mathbb{Q}/\mathbb{Z})$ for $A_0 \subset A$ a finitely generated (and necessarily torsion) subgroup. 
In fact, this gives an anti-equivalence between profinite abelian groups and torsion abelian groups, which is sometimes useful (it means, for instance, that a filtered inverse limit of profinite abelian groups is always exact, since the corresponding fact for filtered direct limits  is true in abelian groups).