Direct sum of topological groups

In mathematics, a topological group $G$ is called the topological direct sum of two subgroups $H_{1}$ and $H_{2}$ if the map

$${\begin{aligned}H_{1}\times H_{2}&\longrightarrow G\\(h_{1},h_{2})&\longmapsto h_{1}h_{2}\end{aligned}}$$

is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

Definition

More generally, $G$ is called the direct sum of a finite set of subgroups $H_{1},\ldots ,H_{n}$ of the map

$${\begin{aligned}\prod _{i=1}^{n}H_{i}&\longrightarrow G\\(h_{i})_{i\in I}&\longmapsto h_{1}h_{2}\cdots h_{n}\end{aligned}}$$

is a topological isomorphism.

If a topological group $G$ is the topological direct sum of the family of subgroups $H_{1},\ldots ,H_{n}$ then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family $H_{i}.$

Topological direct summands

Given a topological group $G,$ we say that a subgroup $H$ is a topological direct summand of $G$ (or that splits topologically from $G$) if and only if there exist another subgroup $K\leq G$ such that $G$ is the direct sum of the subgroups $H$ and $K.$

A the subgroup $H$ is a topological direct summand if and only if the extension of topological groups

$$0\to H{\stackrel {i}{{}\to {}}}G{\stackrel {\pi }{{}\to {}}}G/H\to 0$$

splits, where $i$ is the natural inclusion and $\pi$ is the natural projection.

Examples

Suppose that $G$ is a locally compact abelian group that contains the unit circle $\mathbb {T}$ as a subgroup. Then $\mathbb {T}$ is a topological direct summand of $G.$ The same assertion is true for the real numbers $\mathbb {R}$

See also

• Complemented subspace
• Direct sum – Operation in abstract algebra composing objects into "more complicated" objects
• Direct sum of modules – Operation in abstract algebra