---
tag: QDWNHEQJRK
kind: statement
role: theorem
name: Structure theorem for compactly generated LCA groups
msc: [22B05]
uses: [ZR09S9R3AW, 0RY0B72XAS]
about: [ZR09S9R3AW]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_compact_abelian_group
---

## Statement

Let $G$ be a [[locally compact abelian group|ZR09S9R3AW]].
Assume that $G$ is compactly generated, that is, there is a compact subset $C \subseteq G$ such that $G$ is the smallest subgroup of $G$ containing $C$.
Then there are integers $n, m \ge 0$ and a compact abelian group $K$ such that $G$ is isomorphic, as a topological group, to $\R^n \times \Z^m \times K$.

## Remarks

For a discrete group, compactly generated means finitely generated, and the theorem gives the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]] 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.

## Explanations

### L1 human

Any locally compact abelian group that is generated by a compact piece is built from three simple kinds of blocks.
There are copies of the real line, copies of the integers, and one compact group.
This reduces many questions about such groups to these three cases.

### L2 human

Some proofs of [[Pontryagin duality|QTVC6F2P9D]] 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.