E1 - cosets

introduction

coset

consider a group $G$ , with a non-empty subset $H$ , then for any $a \in G$ , the set ${a h ∣ h \in H}$ is denoted by $a H$ , and analogously, $H a = {h a ∣ h \in H}$ and $a H a^{- 1} = {a h a^{- 1} ∣ h \in H}$
when $H$ is a subgroup of $G$ , $a H$ is called the left coset, and $H a$ the right coset of $H$ in $G$ containing $a$

\begin{aligned} (1) H & = H \\ (12) H & = {(12), (12) (13)} = {(12), (132)} = (132) H \\ (13) H & = {(13), (1)} = H \\ (23) H & = {(23), (23) (13)} = {(23), (123)} = (123) H \end{aligned}

here, $| S_{3} | = 6$ and $| H | = 2$ , so there are $6 / 3 = 2$ left cosets, each with two elements

to find the cosets of $H = {1, 15}$ in $G = U (32)$
the first coset is $H = {1, 15}$
for the second, choose an element of $U (32) \notin H$ , eg: 3, so $3 H = {3, 13}$
for the next, chose another element that has not appeared in the previous two cosets, so $5 H = {5, 11}$
and similarly, all elements from $U (32)$ that have not appeared in previous cosets can be picked as representatives until $U (32)$ is exhausted

let $H$ be a subgroup of $G$ , and let $a b, b \in G$
- $a \in a H$
- $a H = H ⟺ a \in H$ , ie. $H$ absorbs an element if and only if the element belongs to $H$
- $(a b) H = a (b H)$ and $H (a b) = (H a) b$
- $a H = b H ⟺ a \in b H$
- $a H = b H$ or $a H \cap b H = Φ$ , ie. two left cosets of $H$ are either identical or disjoint, and any element of a left coset can be used to represent the coset
- $a H = b H ⟺ a^{- 1} b \in H$
- $| a H | = | b H |$ , ie. all left cosets of $H$ have the same size
- $a H = H a ⟺ H = a H a^{- 1}$
- $a H$ is a subgroup of $G$ if and only if $a \in H$
properties 1, 5 and 7 guarantee that the left cosets of $H$ partition $G$ into blocks of equal size
cosets allow organisation of group elements in a coherent way with every element sharing a special property
$H$ is often chosen such that the cosets partition the group in some highly desirable way
eg: if $G$ is a 3-space $R^{3}$ , and $H$ is a plane through the origin, then the coset $(a, b, c) + H$ (component-wise addition) is the plane passing through $(a, b, c)$ , parallel to $H$
cosets of $H$ constitute a partition of $R^{3}$ into planes parallel to $H$
if $G = G L (2, R)$ and $H = S L (2, R)$ , then for any $A \in G$ , $A H$ is the set of all $2 \times 2$ matrices with the same determinant as $A$