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 $$\begin{array}{rl}{H}_1\times H_2& ⟶G\\ (h_1,h_2)& ⟼h_1{h}_2\end{array}$$ 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 $H_1,\dots ,H_n$ of the map $$\begin{array}{rl}\prod _{i=1}^n{H}_i& ⟶G\\ (h_i{)}_{i\in I}& ⟼h_1{h}_2\cdots h_n\end{array}$$ is a topological isomorphism. If a topological group $G$ is the topological direct sum of the family of subgroups $H_1,\dots ,H_n$ 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 $K\le G$ 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