39 lines
1 KiB
Markdown
39 lines
1 KiB
Markdown
---
|
|
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.
|