quotient group

quotient group

Before defining

quotient groups

, some preliminary

definitions

must be introduced and a few established.

Given a group

G

and a

subgroup

H

of

G

, the

http://planetmath.org/node/122

relation

∼

L

on

G

defined by

a

∼

L

b

if and only
if

b

-

1

⁢

a

∈

H

is called

left

congruence

modulo

H

; similarly the relation defined by

a

∼

R

b

if
and only if

a

⁢

b

-

1

∈

H

is called

congruence modulo

H

(observe that these two relations coincide if

G

is

abelian

).

Proposition

.

Left (resp. right) congruence modulo

H

is an

equivalence relation

on

G

.

Proof.

We will only give the proof for left congruence modulo

H

, as the for right congruence modulo

H

is analogous.
Given

a

∈

G

, because

H

is a subgroup,

H

contains the

identity

e

of

G

, so that

a

-

1

⁢

a

=

e

∈

H

; thus

a

∼

L

a

, so

∼

L

is

http://planetmath.org/node/1644

reflexive

. If

b

∈

G

satisfies

a

∼

L

b

, so that

b

-

1

⁢

a

∈

H

, then by the of

H

under the formation of

inverses

,

a

-

1

⁢

b

=

(

b

-

1

⁢

a

)

-

1

∈

H

, and

b

∼

L

a

; thus

∼

L

is

symmetric

. Finally, if

c

∈

G

,

a

∼

L

b

, and

b

∼

L

c

, then we have

b

-

1

⁢

a

,

c

-

1

⁢

b

∈

H

, and the

closure

of

H

under the

binary operation

of

G

gives

c

-

1

⁢

a

=

(

c

-

1

⁢

b

)

⁢

(

b

-

1

⁢

a

)

∈

H

, so that

a

∼

L

c

, from which it follows that

∼

L

is

http://planetmath.org/node/1669

transitive

, hence an equivalence relation.
∎

It follows from the preceding that

G

is partitioned into

mutually disjoint

, non-empty

equivalence
classes

by left (resp. right) congruence modulo

H

, where

a

,

b

∈

G

are in the same equivalence class if and only if

a

∼

L

b

(resp.

a

∼

R

b

); focusing on left congruence modulo

H

, if we denote by

a

¯

the equivalence class containing

a

under

∼

L

, we see that

a

¯

=

{

b

∈

G

∣

b

∼

L

a

}

=

{

b

∈

G

∣

a

-

1

⁢

b

∈

H

}

=

{

b

∈

G

∣

b

=

a

⁢

h

⁢

for some

⁢

h

∈

H

}

=

{

a

⁢

h

∣

h

∈

H

}

⁢

.

Thus the equivalence class under

∼

L

containing

a

is simply the

left coset

a

⁢

H

of

H

in

G

. Similarly the equivalence class under

∼

R

containing

a

is the

right coset

H

⁢

a

of

H

in

G

(when the binary operation of

G

is written additively, our notation for left and right cosets becomes

a

+

H

=

{

a

+

h

∣

h

∈

H

}

and

H

+

a

=

{

h

+

a

∣

h

∈

H

}

). Observe that the equivalence class under either

∼

L

or

∼

R

containing

e

is

e

⁢

H

=

H

. The

index

of

H

in

G

, denoted by

|

G

:

H

|

, is
the

cardinality

of the set

G

/

H

(read “

G

modulo

H

” or just “

G

mod

H

”) of left cosets of

H

in

G

(in fact, one may demonstrate the

existence

of a

bijection

between the set of left cosets of

H

in

G

and the set of right cosets of

H

in

G

, so that we may well take

|

G

:

H

|

to be the cardinality of the set of right cosets of

H

in

G

).

We now attempt to impose a group on

G

/

H

by taking the of the left cosets containing the

elements

a

and

b

, respectively, to be the left coset containing the element

a

⁢

b

; however, because this definition requires a choice of left coset representatives, there is no guarantee that it will yield a

well-defined

binary operation on

G

/

H

. For the of left coset to be well-defined, we must be sure that if

a

′

⁢

H

=

a

⁢

H

and

b

′

⁢

H

=

b

⁢

H

, i.e., if

a

′

∈

a

⁢

H

and

b

′

∈

b

⁢

H

, then

a

′

⁢

b

′

⁢

H

=

a

⁢

b

⁢

H

, i.e., that

a

′

⁢

b

′

∈

a

⁢

b

⁢

H

. Precisely what must be required of the subgroup

H

to ensure the of the above condition is the

content

of the following :

Proposition

.

The rule

(

a

⁢

H

,

b

⁢

H

)

↦

a

⁢

b

⁢

H

gives a well-defined binary operation on

G

/

H

if and only if

H

is a

normal subgroup

of

G

.

Proof.

Suppose first that of left cosets is well-defined by the given rule, i.e, that given

a

′

∈

a

⁢

H

and

b

′

∈

b

⁢

H

, we have

a

′

⁢

b

′

⁢

H

=

a

⁢

b

⁢

H

, and let

g

∈

G

and

h

∈

H

. Putting

a

=

1

,

a

′

=

h

, and

b

=

b

′

=

g

-

1

, our

hypothesis

gives

h

⁢

g

-

1

⁢

H

=

e

⁢

g

-

1

⁢

H

=

g

-

1

⁢

H

; this implies that

h

⁢

g

-

1

∈

g

-

1

⁢

H

, hence that

h

⁢

g

-

1

=

g

-

1

⁢

h

′

for some

h

′

∈

H

. on the left by

g

gives

g

⁢

h

⁢

g

-

1

=

h

′

∈

H

, and because

g

and

h

were chosen arbitrarily, we may conclude that

g

⁢

H

⁢

g

-

1

⊆

H

for all

g

∈

G

, from which it follows that

H

⊴

G

.

Conversely

, suppose

H

is normal in

G

and let

a

′

∈

a

⁢

H

and

b

′

∈

b

⁢

H

. There exist

h

1

,

h

2

∈

H

such that

a

′

=

a

⁢

h

1

and

b

′

=

b

⁢

h

1

; now, we have

a

′

⁢

b

′

=

a

⁢

h

1

⁢

b

⁢

h

2

=

a

⁢

(

b

⁢

b

-

1

)

⁢

h

1

⁢

b

⁢

h

2

=

a

⁢

b

⁢

(

b

-

1

⁢

h

1

⁢

b

)

⁢

h

2

⁢

,

and because

b

-

1

⁢

h

1

⁢

b

∈

H

by

assumption

, we see that

a

′

⁢

b

′

=

a

⁢

b

⁢

h

, where

h

=

(

b

-

1

⁢

h

⁢

b

)

⁢

h

2

∈

H

by the closure of

H

under in

G

. Thus

a

′

⁢

b

′

∈

a

⁢

b

⁢

H

, and because left cosets
are either

disjoint

or equal, we may conclude that

a

′

⁢

b

′

⁢

H

=

a

⁢

b

⁢

H

, so that

multiplication

of left cosets is indeed a well-defined binary operation on

G

/

H

.
∎

The set

G

/

H

, where

H

is a normal subgroup of

G

, is readily seen to form a group under the well-defined
binary operation of left coset multiplication (the of each group follows from that of

G

), and is called a

quotient

or

factor group

(more specifically
the

quotient of

G

by

H

). We conclude with several examples of specific quotient groups.

Example

.

A standard example of a quotient group is

ℤ

/

n

⁢

ℤ

, the quotient of the of integers by the

cyclic subgroup

generated by

n

∈

ℤ

+

; the order of

ℤ

/

n

⁢

ℤ

is

n

, and the
distinct left cosets of the group are

n

⁢

ℤ

,

1

+

n

⁢

ℤ

,

…

,

(

n

-

1

)

+

n

⁢

ℤ

.

Example

.

Although the group

Q

8

is not abelian, each of its subgroups its normal, so any will suffice for the formation
of quotient groups; the quotient

Q

8

/

⟨

-

1

⟩

, where

⟨

-

1

⟩

=

{

1

,

-

1

}

is the cyclic subgroup of

Q

8

generated by

-

1

, is of order

4

, with elements

⟨

-

1

⟩

,

i

⁢

⟨

-

1

⟩

=

{

i

,

-

i

}

,

k

⁢

⟨

-

1

⟩

=

{

k

,

-

k

}

, and

j

⁢

⟨

-

1

⟩

=

{

j

,

-

j

}

. Since each non-identity element of

Q

8

/

⟨

-

1

⟩

is of order

2

, it is

isomorphic

to the Klein

4

-group

V

. Because each of

⟨

i

⟩

,

⟨

j

⟩

, and

⟨

k

⟩

has order

4

, the quotient of

Q

8

by any of these subgroups is necessarily cyclic of order

2

.

Example

.

The center of the

dihedral group

D

6

of order

12

(with

http://planetmath.org/node/2182

presentation

⟨

r

,

s

∣

r

6

=

s

2

=

1

,

r

-

1

⁢

s

=

s

⁢

r

⟩

) is

⟨

r

3

⟩

=

{

1

,

r

3

}

; the elements of the quotient

D

6

/

⟨

r

3

⟩

are

⟨

r

3

⟩

,

r

⁢

⟨

r

3

⟩

=

{

r

,

r

4

}

,

r

2

⁢

⟨

r

3

⟩

=

{

r

2

,

r

5

}

,

s

⁢

⟨

r

3

⟩

=

{

s

,

s

⁢

r

3

}

,

s

⁢

r

⁢

⟨

r

3

⟩

=

{

s

⁢

r

,

s

⁢

r

4

}

, and

s

⁢

r

2

⁢

⟨

r

3

⟩

=

{

s

⁢

r

2

,

s

⁢

r

5

}

; because

s

⁢

r

2

⁢

⟨

r

3

⟩

⁢

r

⁢

⟨

r

3

⟩

=

s

⁢

r

3

⁢

⟨

r

3

⟩

=

s

⁢

⟨

r

3

⟩

≠

s

⁢

r

⁢

⟨

r

3

⟩

=

r

⁢

⟨

r

3

⟩

⁢

s

⁢

r

2

⁢

⟨

r

3

⟩

⁢

,

D

6

/

⟨

r

3

⟩

is

non-abelian

, hence must be isomorphic to

S

3

.

Title

quotient group

Canonical

name

QuotientGroup

Date of creation

2013-03-22 12:04:06

Last modified on

2013-03-22 12:04:06

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

35

Author

azdbacks4234 (14155)

Entry type

Definition

Classification

msc 20-00

Synonym

factor group

Synonym

quotient

Related topic

Group

Related topic

NormalSubgroup

Related topic

Subgroup

Related topic

EquivalenceRelation

Related topic

Coset

Related topic

NaturalProjection

Defines

left congruence modulo a subgroup

Defines

right congruence modulo a subgroup

Defines

index

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