group theory

# Contents

## Definition

If $G$ is a topological group, then the identity component is the connected component of the identity element $e$ in $G$.

## Basic results

###### Proposition

The identity component ${G}_{0}$ is a closed normal subgroup of $G$.

###### Proof

It is clearly closed (indeed, any connected component is closed). If $g,h\in {G}_{0}$, then $gh$ is in the same connected component as $g$ (since $h$ is in the same connected component as $e$ and left multiplication by $g$ is a homeomorphism), which in turn is in the same connected component as $e$. Using similar reasoning, if $g$ is in the connected component as $e$, then $e$ is in the same connected component as ${g}^{-1}$. Hence ${G}_{0}$ is a subgroup.

If $\varphi$ is any automorphism of $G$, then $\varphi \left({G}_{0}\right)={G}_{0}$. (Indeed, $\varphi \left({G}_{0}\right)$ is a connected set containing $e$ and therefore $\varphi \left({G}_{0}\right)\subseteq {G}_{0}$. Replacing $\varphi$ by its inverse ${\varphi }^{-1}$, we similarly have ${\varphi }^{-1}\left({G}_{0}\right)\subseteq {G}_{0}$ and therefore ${G}_{0}\subseteq \varphi \left({G}_{0}\right)$.) Applying this to inner automorphisms $\varphi$, we conclude that ${G}_{0}$ is a normal subgroup of $G$.

• Remark: ${G}_{0}$ need not be open in $G$; for example, for the group of $p$-adic integers, ${G}_{0}$ is the (non-open) singleton $\left\{e\right\}$. However, if $G$ is locally connected, for example if $G$ is a Lie group, then ${G}_{0}$ is open (and therefore also clopen. In this case, $G/{G}_{0}$ is discrete (because $G\to G/{G}_{0}$ is an open map, implying that the identity and therefore every point in $G/{G}_{0}$ is open).
###### Proposition

The group $G/{G}_{0}$, equipped with the quotient space topology, is a Hausdorff topological group.

###### Proof

Given the fact that $p:G\to G/{G}_{0}$ is an open surjection, the product $p×p:G×G\to G/{G}_{0}×G/{G}_{0}$ is also an open surjection and therefore a quotient map. It follows easily from the universal property of quotient maps that the multiplication $G×G\to G$ therefore descends to a continuous multiplication $G/{G}_{0}×G/{G}_{0}\to G/{G}_{0}$, so that $G/{G}_{0}$ is a topological group.

Because a topological group is a uniform space, the Hausdorff condition follows from a weaker separation axiom such as ${T}_{1}$ (points are closed). It suffices that the identity of $G/{G}_{0}$ be closed. Its complement $C$ is the image under $p$ of the complement of ${G}_{0}$ in $G$ (just by examining coset decompositions), which is open. Since $p$ is an open map, it follows that $C$ is open, so that $\left\{e\right\}$ is closed, as desired.

Created on February 3, 2012 23:46:11 by Todd Trimble (74.88.146.52)