8.Coset

Overview

Given the equivalence relation $R$on A set $A$, we can divide $A$into several equivalence classes by $R$. These equivalence classes form a partition of a set.

Conversely, given a partition of a set, we can define an equivalence relation: if two elements belong to the same subset, then they are equivalent. This gives us an equivalence relation.

Suppose ${\pi }_0$ is a plane passing through the origin $O$, and $A,B$ are any two points in space. Establish a binary relation

$$R=\{⟨\overrightarrow{OA},\overrightarrow{OB}⟩\mid \overrightarrow{OA}-\overrightarrow{OB}\in {\pi }_0\}$$

It is clear that $R$ is an equivalence relation. Each equivalence class is parallel (or coincident) with the plane ${\pi }_0$. Thus, ${\pi }_0$ and all planes parallel to ${\pi }_0$ form a partition of the space. From the perspective of coordinate systems, the vector addition operation for all points in space can constitute a group. $\overrightarrow{OA}-\overrightarrow{OB}$ is equivalent to $\overrightarrow{OA}+{\overrightarrow{OB}}^{-1}$.

Definition

Cosets are equivalence classes.

Relation $\sim$ classification: For $\mathrm{\forall }a\in G$, $[a{]}_{\sim }=\{x\in G,x\sim a\}=\{x\mid x\in G,x{a}^{-1}\in H\}=\{x\mid x=ah\}$, at this time $[a{]}_{\sim }$ can be denoted as $Ha$, called the right coset.

Similarly, if another relation $\sim$ is defined as: $a\sim b\iff b^{-1}a\in H$, then $\sim$ is also an equivalence relation. For $\mathrm{\forall }a\in G$, $[a{]}_{\sim }=\{ah\mid h\in H\}=aH$, called the left coset.

Suppose $<H,\ast >$ is a subgroup of $<G,\ast >$, $a$ is any element in $G$, then

(1) $aH=\{ah\mid h\in H\}$ is a left coset of subgroup $H$ in group $G$;

(2) $Ha=\{ha\mid h\in H\}$ is a right coset of subgroup $H$ in group $G$.

Here, $a$ is called the representative of the left coset $aH$ (or right coset $Ha$). Obviously, when $G$ is an abelian group, the left and right cosets of subgroup $H$​ are equal.

Intuitive understanding

All cosets of a subgroup together form a group, and a coset is a group partition.

Different representative elements are actually divided in different ways, such as $a$ and $b$ in the figure above.

Example

Calculate all the left and right cosets of the subgroup $<\{0,2,4\},\oplus >$ in the group $<{\mathbb{Z}}_6,\oplus >$.

Let $H=\{0,2,4\}$, then all right cosets are:

$H0=\{0,2,4\}0=\{0,2,4\}$,

$H1=\{0,2,4\}1=\{1,3,5\}$,

$H2=\{0,2,4\}2=\{2,4,0\}$,

$H3=\{0,2,4\}3=\{3,5,1\}$,

$H4=\{0,2,4\}4=\{4,0,2\}$,

$H5=\{0,2,4\}5=\{5,1,3\}$.

Thus: $H0=H2=H4$, $H1=H3=H5$, $H0\cup H1={\mathbb{Z}}_6$;

Similarly, all left cosets are:

$0H=0\{0,2,4\}=\{0,2,4\}$,

$1H=1\{0,2,4\}=\{1,3,5\}$,

$2H=2\{0,2,4\}=\{2,4,0\}$,

$3H=3\{0,2,4\}=\{3,5,1\}$,

$4H=4\{0,2,4\}=\{4,0,2\}$,

$5H=5\{0,2,4\}=\{5,1,3\}$.

Thus: $0H=2H=4H$, $1H=3H=5H$, $0H\cup 1H={\mathbb{Z}}_6$.

Property

Let $< H, * > $ be a subgroup of $<G,\ast >$, with $e$ as the identity element, $a,b\in G$, then:

(1) $eH=H=He$;

(2) $Ha=H\iff a\in H(aH=H\iff a\in H)$;

(3) $a\in Hb\iff Ha=Hb\iff a{b}^{-1}\in H(a\in bH\iff aH=bH\iff a^{-1}b\in H)$.

(4) The set of all left cosets (or right cosets) of $H$ forms a partition of $G$.

Let $H$ be a subgroup of group $G$, then for $\mathrm{\forall }a\in G$, $|aH|=|Ha|=|H|$.

Proof:

First prove $|H|=|Ha|$.

Define the mapping $f:H\to Ha$, for $\mathrm{\forall }h\in H$, $f(h)=ha$.

Now prove $f$ is injective:

For $\mathrm{\forall }{h}_1,h_2\in H$, if $h_1\ne h_2$, then $h_{1}a\ne h_{2}a$ (otherwise, by cancellation law, $h_1=h_2$, contradiction), thus $f(h_1)\ne f(h_2)$, so $f$ is injective.

Also, since $Ha=\{ha\mid h\in H\}$, for $\mathrm{\forall }ha\in Ha$, there exists $f(h)=ha$, so $f$ is surjective.

Therefore, $f$ is bijective, thus $|H|=|Ha|$. Similarly prove: $|H|=|aH|$,

i.e., $|Ha|=|H|=|aH|$.

To prove that the number is the same, we only need to prove that there is a bijective relation, and to prove that there is a bijective only need to prove that there is a monojective and a surjective.