---
title: groups
tags: maths, groups
---

# Quotient groups

Wiki: https://en.wikipedia.org/wiki/Coset

`Normal` subgroup of **G** is any subgroup of **G** which is closed with respect to conjugates.

### Theorem 1

If $H$ is a normal subgroup of $G$, then $aH = Ha$ for every $a \in G$.

There is no distinction between left and right cosets for a normal subgroup.

### Theorem 2

Let $H$ be a normal subgroup of $G$. If $Ha = Hc$ and $Hb = Hd$, then $H(ab) = H(cd)$.

### Theorem 3

$G/H$ with coset multiplication is a group.

### Theorem 4

$G/H$ is a homomorphic image of $G$.

### Theorem 5

Let $G$ be a group and $H$ a subgroup of $G$. Then

- $Ha = Hb$  _iff_ $ab^{-1} \in H$ and
- $Ha = H$ _iff_ $a \in H$

The motive for the quotient group construction is that it gives a way of actually _producing_ all the homomorphic images of any group $G$.

For quotient group construction in practical instances, $H$ is often chosen to "factor out" unwanted properties of $G$, and preserve in $G/H$ only "desirable" traits.