Cosets

Introduction

A subgroup $H$ of a group $G$ provides a natural way to partition the elements of $G$ into equal-sized pieces called cosets. This partitioning is fundamental to understanding the structure of groups.

Definition

Definition 2.1: Let $H$ be a subgroup of a group $G$ , and let $g \in G$ .

The left coset of $H$ with respect to $g$ is the set $g H = {g h ∣ h \in H}$
The right coset of $H$ with respect to $g$ is the set $H g = {h g ∣ h \in H}$
The element $g$ is called a coset representative

Key Properties of Cosets

1. Partition Property

The left cosets of $H$ in $G$ form a partition of the set $G$ . This means that:

Every element of $G$ belongs to exactly one left coset
Any two left cosets are either identical or disjoint

2. Equivalence Relation

This can be proven formally by showing that the relation $a \sim b$ if and only if $a^{- 1} b \in H$ is an equivalence relation on $G$ , where the equivalence classes are precisely the left cosets of $H$ .

3. Equal Cardinality

There is a simple bijection between any subgroup $H$ and any of its cosets $g H$ (given by the map $h \mapsto g h$ ). This implies that all cosets of $H$ have the same cardinality as $H$ itself.

Examples

Example 1: Cosets in $D_{4}$

Consider the dihedral group $D_{4}$ and the subgroup $H = {e, r^{2}}$ (rotations by $0 °$ and $180 °$ ):

$e H = {e \cdot e, e \cdot r^{2}} = {e, r^{2}}$
$r H = {r \cdot e, r \cdot r^{2}} = {r, r^{3}}$
$s H = {s \cdot e, s \cdot r^{2}} = {s, s r^{2}}$
$s r H = {s r \cdot e, s r \cdot r^{2}} = {s r, s r^{3}}$

These four cosets partition $D_{4}$ into equal-sized pieces, each containing 2 elements.

Example 2: Cosets in $(Z, +)$

Consider the subgroup $H = 3 Z = {0, \pm 3, \pm 6, \dots}$ in $(Z, +)$ :

$0 + H = {0, \pm 3, \pm 6, \dots} = H$
$1 + H = {1, 4, 7, \dots, - 2, - 5, \dots}$
$2 + H = {2, 5, 8, \dots, - 1, - 4, \dots}$

These three cosets partition $Z$ into three infinite sets.

Example 3: Cosets in $S_{3}$

Consider the subgroup $H = {e, (1, 2)}$ in $S_{3}$ :

$e H = {e, (1, 2)}$
$(1, 3) H = {(1, 3), (1, 2, 3)}$
$(2, 3) H = {(2, 3), (1, 3, 2)}$

These three cosets partition $S_{3}$ into equal-sized pieces.

Properties

Coset Representatives

Any element of a coset can serve as a coset representative. If $g^{'} \in g H$ , then $g^{'} H = g H$ .

Left vs Right Cosets

In general, left cosets and right cosets may be different. However, they coincide if and only if the subgroup is normal.

Number of Cosets

The number of left cosets equals the number of right cosets, and this number is called the index of $H$ in $G$ , denoted $[G : H]$ .

Applications

Application 1: Group Partitioning

Cosets provide a systematic way to partition a group into equal-sized pieces.

Application 2: Index Calculations

The index $[G : H]$ can be calculated as $[G : H] = | G | / | H |$ for finite groups.

Application 3: Normal Subgroups

The study of when left and right cosets coincide leads to the concept of normal subgroups.