Cosets as Equivalence Classes

Cosets can also be understood through equivalence relations. This viewpoint explains why cosets form partitions: every equivalence relation partitions a set into equivalence classes. For a subgroup

H

of a group

G

, the relation

a\rho b

defined by

a^{-1}\circ b\in H

produces left cosets. A related relation defined by

b\circ a^{-1}\in H

produces right cosets. This lecture connects the algebraic definition of a coset with the general set-theoretic idea of equivalence classes.

Let

(G,\circ)

be a group and let

H

be a subgroup of

G

. The

\textbf{left coset relation}

G

determined by

H

is the relation

\rho_L

defined by

\begin{equation*} a\rho_L b \iff a^{-1}\circ b\in H \end{equation*}

for all

a,b\in G

Cosets can also be understood through equivalence relations. This viewpoint explains why cosets form partitions: every equivalence relation partitions a set into equivalence classes. For a subgroup

H

of a group

G

, the relation

a\rho b

defined by

a^{-1}\circ b\in H

produces left cosets. A related relation defined by

b\circ a^{-1}\in H

produces right cosets. This lecture connects the algebraic definition of a coset with the general set-theoretic idea of equivalence classes.

Let

(G,\circ)

be a group and let

H

be a subgroup of

G

. The

\textbf{left coset relation}

G

determined by

H

is the relation

\rho_L

defined by

\begin{equation*} a\rho_L b \iff a^{-1}\circ b\in H \end{equation*}

for all

a,b\in G

Let

(G,\circ)

be a group and let

H

be a subgroup of

G

. Let

\rho_L

be the relation on

G

defined by

\begin{equation*} a\rho_L b \iff a^{-1}\circ b\in H. \end{equation*}

Then

\rho_L

is an equivalence relation and

\begin{equation*} \operatorname{cl}(a)=aH \end{equation*}

for every

a\in G

Given that

(G,\circ)

is a group,

H

is a subgroup of

G

, and

\rho_L

is defined on

G

\begin{equation*} a\rho_L b \iff a^{-1}\circ b\in H. \end{equation*}

To prove that

\rho_L

is an equivalence relation and

\operatorname{cl}(a)=aH

for every

a\in G

. [1] To prove reflexivity. Let

a\in G

. Then

\begin{align*} a^{-1}\circ a &=e\\ &\in H. \end{align*}

Therefore

a\rho_L a

. [2] To prove symmetry. Let

a,b\in G

and let

a\rho_L b

. Then

\begin{equation*} a^{-1}\circ b\in H. \end{equation*}

Since

H

is a subgroup, it is closed under inverses. Therefore

\begin{align*} a^{-1}\circ b\in H &\implies (a^{-1}\circ b)^{-1}\in H\\ &\implies b^{-1}\circ a\in H. \end{align*}

Therefore

b\rho_L a

. [3] To prove transitivity. Let

a,b,c\in G

such that

a\rho_L b

and

b\rho_L c

. Then

\begin{equation*} a^{-1}\circ b\in H \end{equation*}

and

\begin{equation*} b^{-1}\circ c\in H. \end{equation*}

Since

H

is closed under

\circ

, we get

\begin{align*} (a^{-1}\circ b)\circ (b^{-1}\circ c)\in H &\implies a^{-1}\circ c\in H. \end{align*}

Therefore

a\rho_L c

. Therefore

\rho_L

is an equivalence relation. Now let

a\in G

. Then

\begin{align*} \operatorname{cl}(a) &=\{b\in G:a\rho_L b\}\\ &=\{b\in G:a^{-1}\circ b\in H\}\\ &=\{b\in G:b\in aH\}\\ &=aH. \end{align*}

Hence,

\rho_L

is an equivalence relation and

\operatorname{cl}(a)=aH

for every

a\in G

\square

We observe the equivalence class of an element under the left coset relation in

\mathbb{Z}_n

. Choose an element

a

and the preview finds all

b

such that

-a+b

lies in

H

. The resulting class is exactly the coset

a+H

. This connects the algebraic translation picture with the set-theoretic idea of equivalence classes.

Let

(G,\circ)

be a group and let

H

be a subgroup of

G

. The

\textbf{right coset relation}

G

determined by

H

is the relation

\rho_R

defined by

\begin{equation*} a\rho_R b \iff b\circ a^{-1}\in H \end{equation*}

for all

a,b\in G

Let

(G,\circ)

be a group and let

H

be a subgroup of

G

. Let

\rho_R

be the relation on

G

defined by

\begin{equation*} a\rho_R b \iff b\circ a^{-1}\in H. \end{equation*}

Then

\rho_R

is an equivalence relation and

\begin{equation*} \operatorname{cl}(a)=Ha \end{equation*}

for every

a\in G

Given that

(G,\circ)

is a group,

H

is a subgroup of

G

, and

\rho_R

is defined on

G

\begin{equation*} a\rho_R b \iff b\circ a^{-1}\in H. \end{equation*}

To prove that

\rho_R

is an equivalence relation and

\operatorname{cl}(a)=Ha

for every

a\in G

. [1] To prove reflexivity. Let

a\in G

. Then

\begin{align*} a\circ a^{-1} &=e\\ &\in H. \end{align*}

Therefore

a\rho_R a

. [2] To prove symmetry. Let

a,b\in G

and let

a\rho_R b

. Then

\begin{equation*} b\circ a^{-1}\in H. \end{equation*}

Since

H

is a subgroup, it is closed under inverses. Therefore

\begin{align*} b\circ a^{-1}\in H &\implies (b\circ a^{-1})^{-1}\in H\\ &\implies a\circ b^{-1}\in H. \end{align*}

Therefore

b\rho_R a

. [3] To prove transitivity. Let

a,b,c\in G

such that

a\rho_R b

and

b\rho_R c

. Then

\begin{equation*} b\circ a^{-1}\in H \end{equation*}

and

\begin{equation*} c\circ b^{-1}\in H. \end{equation*}

Since

H

is closed under

\circ

, we get

\begin{align*} (c\circ b^{-1})\circ (b\circ a^{-1})\in H &\implies c\circ a^{-1}\in H. \end{align*}

Therefore

a\rho_R c

. Therefore

\rho_R

is an equivalence relation. Now let

a\in G

. Then

\begin{align*} \operatorname{cl}(a) &=\{b\in G:a\rho_R b\}\\ &=\{b\in G:b\circ a^{-1}\in H\}\\ &=\{b\in G:b\in Ha\}\\ &=Ha. \end{align*}

Hence,

\rho_R

is an equivalence relation and

\operatorname{cl}(a)=Ha

for every

a\in G

\square

Let

G=\mathbb{Z}_{12}

under addition modulo

12

and let

\begin{equation*} H=\{0,4,8\}. \end{equation*}

For the left coset relation, in additive notation,

\begin{equation*} a\rho_L b \iff -a+b\in H. \end{equation*}

The class of

1

\begin{align*} \operatorname{cl}(1) &=1+H\\ &=\{1,5,9\}. \end{align*}

Thus the equivalence class of

1

is exactly the coset determined by

1

Let

G=\mathbb{Z}_{10}

under addition modulo

10

and let

H=\{0,5\}

. For the relation

a\rho b

if and only if

-a+b\in H

, find

\operatorname{cl}(3)

Let

G=\mathbb{Z}_{10}

and

H=\{0,5\}

. The relation is the left coset relation in additive notation. Therefore

\begin{align*} \operatorname{cl}(3) &=3+H\\ &=\{3+0,\;3+5\}\\ &=\{3,8\}. \end{align*}

We compute the equivalence class of an element for the modular left coset relation. The calculator checks each residue

b

and keeps exactly those satisfying

-a+b\in H

. It then compares the class with the coset

a+H

. This makes the identity between equivalence classes and cosets visible element by element.

1. Let

G=\mathbb{Z}_{9}

and let

H=\{0,3,6\}

. For the relation

a\rho b

if and only if

-a+b\in H

, find

\operatorname{cl}(2)

. 2. Let

(G,\circ)

be a group and let

H

be a subgroup of

G

. Prove that

a\rho_L b

if and only if

b\in aH

. 3. Let

(G,\circ)

be a group and let

H

be a subgroup of

G

. Prove that

a\rho_R b

if and only if

b\in Ha

1. We have

\begin{align*} \operatorname{cl}(2) &=2+H\\ &=\{2,5,8\}. \end{align*}

2. By definition,

a\rho_L b

if and only if

a^{-1}\circ b\in H

. By the left coset membership criterion, this is equivalent to

b\in aH

. 3. By definition,

a\rho_R b

if and only if

b\circ a^{-1}\in H

. By the right coset membership criterion, this is equivalent to

b\in Ha

Cosets are equivalence classes for natural relations determined by the subgroup.

The relation

a\rho_L b

if and only if

a^{-1}\circ b\in H

gives left cosets.

The relation

a\rho_R b

if and only if

b\circ a^{-1}\in H

gives right cosets.

BMLabs Mathematics

BMLabs Mathematics

Cosets as Equivalence Classes

Cosets as Equivalence Classes

OVERVIEW

LEFT COSET RELATION

EXPLORE A COSET EQUIVALENCE CLASS

COSET EQUIVALENCE CLASS EXPLORER

RIGHT COSET RELATION

EQUIVALENCE CLASSES IN INTEGERS MODULO TWELVE

FINDING AN EQUIVALENCE CLASS

SOLUTION

CALCULATE AN EQUIVALENCE CLASS

EQUIVALENCE CLASS CALCULATOR

EXERCISES

ANSWERS

Coordinates and Locus

Invariants under Orthogonal Transformation