Structure theorem for compactly generated LCA groups

Remarks

For a discrete group, compactly generated means finitely generated, and the theorem gives the structure theorem for finitely generated abelian groups up to the finite part, which goes into $K$. Example: $G = \R \times \Z/2\Z$ is compactly generated by $[0, 1] \times \Z/2\Z$, and it already has the form $\R^1 \times \Z^0 \times K$ with $K = \Z/2\Z$. The hypothesis is needed: the field $\Q_p$ of $p$-adic numbers is locally compact abelian but not compactly generated. It is not of the form $\R^n \times \Z^m \times K$. Indeed each element of $\Q_p$ lies in a compact subgroup $p^{-k}\Z_p$, which would force $n = m = 0$, but $\Q_p$ is not compact. Every locally compact abelian group has an open compactly generated subgroup, so it has an open subgroup isomorphic to $\R^n \times K$ with $K$ compact.

Going deeper

For a student

Some proofs of Pontryagin duality use the theorem: one checks duality on $\R$, $\Z$ and compact groups, and then passes to open subgroups and quotients. It also shows that a connected locally compact abelian group is isomorphic to $\R^n \times K$ with $K$ compact and connected. For non-compactly generated groups, such as $\Q_p$ or the adeles, the theorem applies only to open subgroups, and the global structure can be much richer.