Normal Subgroups and Quotient Groups

We define normal subgroups and establish their equivalent characterizations, then construct the quotient group G/N and prove its basic properties. Topics include the canonical projection, the correspondence between normal subgroups and kernels, subgroups of index 2, and an introduction to simple groups.

Folio Official

March 1, 2026

1 Motivation

When we proved Lagrange's theorem in the previous chapter, we saw that the left cosets of a subgroup $H$ partition the group $G$ . A natural question arises: can we turn this partition into a group?

If we try to define a "product" of cosets by $a H \cdot b H = (ab) H$ , we immediately encounter a problem. If $a H = a^{'} H$ and $b H = b^{'} H$ , it is not automatically true that $(ab) H = (a^{'} b^{'}) H$ . This product is well-defined precisely when $H$ is a normal subgroup.

2 Definition of Normal Subgroups

Definition 1 (Normal subgroup).

A subgroup

N

of a group

G

is called a normal subgroup if

g N = N g

for every

g \in G

. We write

N ⊴ G

. If

N \neq = {e}

and

N \neq = G

, we say

N

is a proper normal subgroup and write

N ◃ G

Remark 2.

The condition

g N = N g

is an equality of sets, not of individual elements. It does not mean that

g n = n g

for every

n \in N

. Rather, it says: for every

n \in N

, there exists

n^{'} \in N

such that

g n = n^{'} g

(and vice versa).

3 Equivalent Conditions for Normality

Theorem 3 (Characterizations of normal subgroups).

Let

G

be a group and

N \leq G

. The following are equivalent:

$N ⊴ G$ (i.e. $g N = N g$ for all $g \in G$ ).
$g N g^{- 1} = N$ for all $g \in G$ .
$g n g^{- 1} \in N$ for all $g \in G$ and $n \in N$ .
$g N g^{- 1} \subseteq N$ for all $g \in G$ .

Proof.

(1) \Rightarrow (2)

: Multiplying

g N = N g

on the right by

g^{- 1}

gives

g N g^{- 1} = N g g^{- 1} = N

(2) \Rightarrow (3)

: If

g N g^{- 1} = N

, then for any

n \in N

we have

g n g^{- 1} \in g N g^{- 1} = N

(3) \Rightarrow (4)

: Immediate.

(4) \Rightarrow (1)

: If

g N g^{- 1} \subseteq N

for all

g

, then

g N \subseteq N g

. Replacing

g

g^{- 1}

gives

g^{- 1} N g \subseteq N

, i.e.

N g \subseteq g N

. Therefore

g N = N g

. □

Definition 4 (Conjugation).

For

g \in G

and

n \in N

, the element

g n g^{- 1}

is called the conjugate of

n

g

. Condition (3) above says that

N

is closed under conjugation by every element of $G$ .

4 Examples of Normal Subgroups

Example 5 (Trivial normal subgroups).

For any group

G

, both

{e} ⊴ G

and

G ⊴ G

Example 6 (Abelian groups).

G

is abelian, then every subgroup of

G

is normal. Indeed,

g n g^{- 1} = g g^{- 1} n = n \in N

for all

g \in G

and

n \in N

Example 7 (Subgroups of index

2

[G : N] = 2

, then

N ⊴ G

Proof.

g \in N

, then

g N = N = N g

trivially. If

g \in / N

, there are only two left cosets:

N

and

g N

. Since

g N \neq = N

, we must have

g N = G ∖ N

. Similarly, the two right cosets are

N

and

N g = G ∖ N

. Therefore

g N = N g

. □

Example 8 (

A_{n} ⊴ S_{n}

The alternating group

A_{n}

is a normal subgroup of

S_{n}

. This follows immediately from the preceding result, since

[S_{n} : A_{n}] = 2

Example 9 (

S L_{n} (R) ⊴ G L_{n} (R)

For

A \in G L_{n} (R)

and

B \in S L_{n} (R)

, we have

det (A B A^{- 1}) = det (A) det (B) det (A)^{- 1} = det (B) = 1

, so

A B A^{- 1} \in S L_{n} (R)

Example 10 (The kernel of a homomorphism is normal).

For any group homomorphism

φ : G \to G^{'}

, we have

ker φ ⊴ G

. Indeed, if

n \in ker φ

, then

φ (g n g^{- 1}) = φ (g) φ (n) φ (g)^{- 1} = φ (g) e_{G^{'}} φ (g)^{- 1} = e_{G^{'}}

, so

g n g^{- 1} \in ker φ

Example 11 (A non-normal subgroup).

S_{3}

, the subgroup

H = {e, (12)}

is not normal. Indeed,

(13) (12) (13)^{- 1} = (13) (12) (13) = (23) \in / H

5 Construction of the Quotient Group

Theorem 12 (Quotient group).

Let

N ⊴ G

. Define

G / N = {g N ∣ g \in G}

(the set of all left cosets) with the operation

(a N) (b N) = (ab) N .

Then

G / N

is a group, called the quotient group (or factor group) of

G

N

Proof.

Well-definedness. Suppose

a N = a^{'} N

and

b N = b^{'} N

. Write

a^{'} = a n_{1}

and

b^{'} = b n_{2}

with

n_{1}, n_{2} \in N

. Then

a^{'} b^{'} = a n_{1} b n_{2} = a (n_{1} b) n_{2}

. Since

N

is normal,

n_{1} b = b n_{1}^{'}

for some

n_{1}^{'} \in N

. Hence

a^{'} b^{'} = ab n_{1}^{'} n_{2}

with

n_{1}^{'} n_{2} \in N

, so

(a^{'} b^{'}) N = (ab) N

Associativity.

(a N \cdot b N) \cdot c N = (ab) N \cdot c N = ((ab) c) N = (a (b c)) N = a N \cdot (b c) N = a N \cdot (b N \cdot c N)

Identity. The identity element is

e N = N

. Indeed,

a N \cdot N = (a e) N = a N

Inverses. The inverse of

a N

a^{- 1} N

. Indeed,

a N \cdot a^{- 1} N = (a a^{- 1}) N = N

. □

Example 13 (

Z / n Z

Since

Z

is abelian,

n Z ⊴ Z

. The quotient group

Z / n Z

is a cyclic group of order

n

, with elements

\overset{ˉ}{0}, \overset{ˉ}{1}, \dots, \overline{n - 1}

and the operation

\overset{a}{ˉ} + \overset{ˉ}{b} = \overline{a + b}

Example 14 (

S_{3} / A_{3}

We have

A_{3} = {e, (123), (132)} ⊴ S_{3}

. The quotient group

S_{3} / A_{3}

has order

2

and is isomorphic to

Z /2 Z

. Its elements are

A_{3}

(the even permutations) and

(12) A_{3}

(the odd permutations).

Example 15 (

G L_{n} (R) / S L_{n} (R)

Since

S L_{n} (R) ⊴ G L_{n} (R)

, the quotient group is well-defined, and in fact

G L_{n} (R) / S L_{n} (R) ≅ R^{\times}

(via the determinant map).

6 The Canonical Projection

Definition 16 (Natural homomorphism).

For

N ⊴ G

, the map

π : G \to G / N

defined by

π (g) = g N

is called the natural homomorphism (or canonical projection).

Theorem 17.

The natural homomorphism

π : G \to G / N

is a surjective homomorphism with

ker π = N

Proof.

π (ab) = (ab) N = (a N) (b N) = π (a) π (b)

, so

π

is a homomorphism. For any

g N \in G / N

π (g) = g N

, so

π

is surjective. Finally,

ker π = {g \in G ∣ g N = N} = {g \in G ∣ g \in N} = N

. □

Remark 18.

This theorem establishes the fundamental correspondence: normal subgroups are precisely the kernels of homomorphisms. Given a normal subgroup

N

, we recover it as

ker π

for the canonical projection

π : G \to G / N

. Conversely, the kernel of any homomorphism

φ

is a normal subgroup (as shown in the examples above).

7 Homomorphisms and Normal Subgroups

Theorem 19 (Subgroups and homomorphisms).

Let

φ : G \to G^{'}

be a group homomorphism. Then:

If $H \leq G$ , then $φ (H) \leq G^{'}$ .
If $H^{'} \leq G^{'}$ , then $φ^{- 1} (H^{'}) \leq G$ .
If $N^{'} ⊴ G^{'}$ , then $φ^{- 1} (N^{'}) ⊴ G$ .
If $φ$ is surjective and $N ⊴ G$ , then $φ (N) ⊴ G^{'}$ .

Here

φ^{- 1} (H^{'}) = {g \in G ∣ φ (g) \in H^{'}}

denotes the preimage.

Proof.

(1) For

a, b \in H

φ (a) φ (b)^{- 1} = φ (a b^{- 1}) \in φ (H)

(2) For

a, b \in φ^{- 1} (H^{'})

φ (a b^{- 1}) = φ (a) φ (b)^{- 1} \in H^{'}

(since

H^{'} \leq G^{'}

), so

a b^{- 1} \in φ^{- 1} (H^{'})

(3) For

g \in G

and

n \in φ^{- 1} (N^{'})

φ (g n g^{- 1}) = φ (g) φ (n) φ (g)^{- 1} \in N^{'}

(since

N^{'} ⊴ G^{'}

), so

g n g^{- 1} \in φ^{- 1} (N^{'})

(4) For

g^{'} \in G^{'}

and

φ (n) \in φ (N)

: since

φ

is surjective,

g^{'} = φ (g)

for some

g \in G

. Then

g^{'} φ (n) g^{' - 1} = φ (g) φ (n) φ (g)^{- 1} = φ (g n g^{- 1}) \in φ (N)

, using

N ⊴ G

. □

8 Simple Groups

Definition 20 (Simple group).

A group

G

with

∣ G ∣ > 1

is called simple if it has no proper normal subgroups, i.e. the only normal subgroups of

G

are

{e}

and

G

itself.

Theorem 21.

Every group of prime order is simple.

Proof.

By Lagrange's theorem, if

∣ G ∣ = p

(prime), the only possible subgroup orders are

1

and

p

. Hence the only subgroups are

{e}

and

G

, so

G

has no proper normal subgroups. □

Theorem 22.

For

n \geq 5

, the alternating group

A_{n}

is simple.

Remark 23.

The proof that

A_{n}

is simple for

n \geq 5

requires a substantial argument, which we omit here. The group

A_{5}

is the smallest non-abelian simple group, with

∣ A_{5} ∣ = 60

Remark 24.

Simple groups play the role of "primes" in group theory. Every finite group can be built up from simple groups (via composition series). The complete classification of finite simple groups is one of the monumental achievements of twentieth-century algebra.

9 Analyzing Structure via Quotient Groups

Theorem 25 (Correspondence theorem (Fourth isomorphism theorem)).

Let

N ⊴ G

. There is an inclusion-preserving bijection between the subgroups of

G

that contain

N

and the subgroups of

G / N

H \mapsto H / N, \overset{ˉ}{H} \mapsto π^{- 1} (\overset{ˉ}{H}),

where

π : G \to G / N

is the canonical projection. Moreover,

H ⊴ G

if and only if

H / N ⊴ G / N

Proof.

N \leq H \leq G

, then

N ⊴ H

(since

N ⊴ G

and

N \leq H

), so

H / N

is a well-defined group, and

H / N \leq G / N

Conversely, if

\overset{ˉ}{H} \leq G / N

, then

π^{- 1} (\overset{ˉ}{H}) = {g \in G ∣ g N \in \overset{ˉ}{H}}

is a subgroup of

G

containing

N

One verifies that these two correspondences are mutual inverses and preserve both inclusion and normality. □

10 Summary and Next Steps

In this chapter we have covered:

The definition of normal subgroups and their equivalent characterizations (closure under conjugation).
Examples: abelian groups, subgroups of index $2$ , kernels, $A_{n}$ , $S L_{n}$ .
Construction of the quotient group $G / N$ and the proof of well-definedness.
The canonical projection $π : G \to G / N$ and the correspondence "normal subgroups $=$ kernels."
Simple groups: definition and examples.
The correspondence theorem.

With normal subgroups and quotient groups in hand, we are ready to prove the central theorems of group theory — the isomorphism theorems (first, second, and third). These are variations on a single principle: "the image of a homomorphism is isomorphic to a quotient group," and they constitute the most powerful tools for analyzing the structure of groups.

Group Theory Algebra Textbook Normal Subgroups Quotient Groups

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Normal Subgroups and Quotient Groups

Folio Official

March 1, 2026

1 Motivation

When we proved Lagrange's theorem in the previous chapter, we saw that the left cosets of a subgroup $H$ partition the group $G$ . A natural question arises: can we turn this partition into a group?

2 Definition of Normal Subgroups

Definition 1 (Normal subgroup).

A subgroup

N

of a group

G

is called a normal subgroup if

g N = N g

for every

g \in G

. We write

N ⊴ G

. If

N \neq = {e}

and

N \neq = G

, we say

N

is a proper normal subgroup and write

N ◃ G

Remark 2.

The condition

g N = N g

is an equality of sets, not of individual elements. It does not mean that

g n = n g

for every

n \in N

. Rather, it says: for every

n \in N

, there exists

n^{'} \in N

such that

g n = n^{'} g

(and vice versa).

3 Equivalent Conditions for Normality

Theorem 3 (Characterizations of normal subgroups).

Let

G

be a group and

N \leq G

. The following are equivalent:

$N ⊴ G$ (i.e. $g N = N g$ for all $g \in G$ ).
$g N g^{- 1} = N$ for all $g \in G$ .
$g n g^{- 1} \in N$ for all $g \in G$ and $n \in N$ .
$g N g^{- 1} \subseteq N$ for all $g \in G$ .

Proof.

(1) \Rightarrow (2)

: Multiplying

g N = N g

on the right by

g^{- 1}

gives

g N g^{- 1} = N g g^{- 1} = N

(2) \Rightarrow (3)

: If

g N g^{- 1} = N

, then for any

n \in N

we have

g n g^{- 1} \in g N g^{- 1} = N

(3) \Rightarrow (4)

: Immediate.

(4) \Rightarrow (1)

: If

g N g^{- 1} \subseteq N

for all

g

, then

g N \subseteq N g

. Replacing

g

g^{- 1}

gives

g^{- 1} N g \subseteq N

, i.e.

N g \subseteq g N

. Therefore

g N = N g

. □

Definition 4 (Conjugation).

For

g \in G

and

n \in N

, the element

g n g^{- 1}

is called the conjugate of

n

g

. Condition (3) above says that

N

is closed under conjugation by every element of $G$ .

4 Examples of Normal Subgroups

Example 5 (Trivial normal subgroups).

For any group

G

, both

{e} ⊴ G

and

G ⊴ G

Example 6 (Abelian groups).

G

is abelian, then every subgroup of

G

is normal. Indeed,

g n g^{- 1} = g g^{- 1} n = n \in N

for all

g \in G

and

n \in N

Example 7 (Subgroups of index

2

[G : N] = 2

, then

N ⊴ G

Proof.

g \in N

, then

g N = N = N g

trivially. If

g \in / N

, there are only two left cosets:

N

and

g N

. Since

g N \neq = N

, we must have

g N = G ∖ N

. Similarly, the two right cosets are

N

and

N g = G ∖ N

. Therefore

g N = N g

. □

Example 8 (

A_{n} ⊴ S_{n}

The alternating group

A_{n}

is a normal subgroup of

S_{n}

. This follows immediately from the preceding result, since

[S_{n} : A_{n}] = 2

Example 9 (

S L_{n} (R) ⊴ G L_{n} (R)

For

A \in G L_{n} (R)

and

B \in S L_{n} (R)

, we have

det (A B A^{- 1}) = det (A) det (B) det (A)^{- 1} = det (B) = 1

, so

A B A^{- 1} \in S L_{n} (R)

Example 10 (The kernel of a homomorphism is normal).

For any group homomorphism

φ : G \to G^{'}

, we have

ker φ ⊴ G

. Indeed, if

n \in ker φ

, then

φ (g n g^{- 1}) = φ (g) φ (n) φ (g)^{- 1} = φ (g) e_{G^{'}} φ (g)^{- 1} = e_{G^{'}}

, so

g n g^{- 1} \in ker φ

Example 11 (A non-normal subgroup).

S_{3}

, the subgroup

H = {e, (12)}

is not normal. Indeed,

(13) (12) (13)^{- 1} = (13) (12) (13) = (23) \in / H

5 Construction of the Quotient Group

Theorem 12 (Quotient group).

Let

N ⊴ G

. Define

G / N = {g N ∣ g \in G}

(the set of all left cosets) with the operation

(a N) (b N) = (ab) N .

Then

G / N

is a group, called the quotient group (or factor group) of

G

N

Proof.

Well-definedness. Suppose

a N = a^{'} N

and

b N = b^{'} N

. Write

a^{'} = a n_{1}

and

b^{'} = b n_{2}

with

n_{1}, n_{2} \in N

. Then

a^{'} b^{'} = a n_{1} b n_{2} = a (n_{1} b) n_{2}

. Since

N

is normal,

n_{1} b = b n_{1}^{'}

for some

n_{1}^{'} \in N

. Hence

a^{'} b^{'} = ab n_{1}^{'} n_{2}

with

n_{1}^{'} n_{2} \in N

, so

(a^{'} b^{'}) N = (ab) N

Associativity.

(a N \cdot b N) \cdot c N = (ab) N \cdot c N = ((ab) c) N = (a (b c)) N = a N \cdot (b c) N = a N \cdot (b N \cdot c N)

Identity. The identity element is

e N = N

. Indeed,

a N \cdot N = (a e) N = a N

Inverses. The inverse of

a N

a^{- 1} N

. Indeed,

a N \cdot a^{- 1} N = (a a^{- 1}) N = N

. □

Example 13 (

Z / n Z

Since

Z

is abelian,

n Z ⊴ Z

. The quotient group

Z / n Z

is a cyclic group of order

n

, with elements

\overset{ˉ}{0}, \overset{ˉ}{1}, \dots, \overline{n - 1}

and the operation

\overset{a}{ˉ} + \overset{ˉ}{b} = \overline{a + b}

Example 14 (

S_{3} / A_{3}

We have

A_{3} = {e, (123), (132)} ⊴ S_{3}

. The quotient group

S_{3} / A_{3}

has order

2

and is isomorphic to

Z /2 Z

. Its elements are

A_{3}

(the even permutations) and

(12) A_{3}

(the odd permutations).

Example 15 (

G L_{n} (R) / S L_{n} (R)

Since

S L_{n} (R) ⊴ G L_{n} (R)

, the quotient group is well-defined, and in fact

G L_{n} (R) / S L_{n} (R) ≅ R^{\times}

(via the determinant map).

6 The Canonical Projection

Definition 16 (Natural homomorphism).

For

N ⊴ G

, the map

π : G \to G / N

defined by

π (g) = g N

is called the natural homomorphism (or canonical projection).

Theorem 17.

The natural homomorphism

π : G \to G / N

is a surjective homomorphism with

ker π = N

Proof.

π (ab) = (ab) N = (a N) (b N) = π (a) π (b)

, so

π

is a homomorphism. For any

g N \in G / N

π (g) = g N

, so

π

is surjective. Finally,

ker π = {g \in G ∣ g N = N} = {g \in G ∣ g \in N} = N

. □

Remark 18.

This theorem establishes the fundamental correspondence: normal subgroups are precisely the kernels of homomorphisms. Given a normal subgroup

N

, we recover it as

ker π

for the canonical projection

π : G \to G / N

. Conversely, the kernel of any homomorphism

φ

is a normal subgroup (as shown in the examples above).

7 Homomorphisms and Normal Subgroups

Theorem 19 (Subgroups and homomorphisms).

Let

φ : G \to G^{'}

be a group homomorphism. Then:

If $H \leq G$ , then $φ (H) \leq G^{'}$ .
If $H^{'} \leq G^{'}$ , then $φ^{- 1} (H^{'}) \leq G$ .
If $N^{'} ⊴ G^{'}$ , then $φ^{- 1} (N^{'}) ⊴ G$ .
If $φ$ is surjective and $N ⊴ G$ , then $φ (N) ⊴ G^{'}$ .

Here

φ^{- 1} (H^{'}) = {g \in G ∣ φ (g) \in H^{'}}

denotes the preimage.

Proof.

(1) For

a, b \in H

φ (a) φ (b)^{- 1} = φ (a b^{- 1}) \in φ (H)

(2) For

a, b \in φ^{- 1} (H^{'})

φ (a b^{- 1}) = φ (a) φ (b)^{- 1} \in H^{'}

(since

H^{'} \leq G^{'}

), so

a b^{- 1} \in φ^{- 1} (H^{'})

(3) For

g \in G

and

n \in φ^{- 1} (N^{'})

φ (g n g^{- 1}) = φ (g) φ (n) φ (g)^{- 1} \in N^{'}

(since

N^{'} ⊴ G^{'}

), so

g n g^{- 1} \in φ^{- 1} (N^{'})

(4) For

g^{'} \in G^{'}

and

φ (n) \in φ (N)

: since

φ

is surjective,

g^{'} = φ (g)

for some

g \in G

. Then

g^{'} φ (n) g^{' - 1} = φ (g) φ (n) φ (g)^{- 1} = φ (g n g^{- 1}) \in φ (N)

, using

N ⊴ G

. □

8 Simple Groups

Definition 20 (Simple group).

A group

G

with

∣ G ∣ > 1

is called simple if it has no proper normal subgroups, i.e. the only normal subgroups of

G

are

{e}

and

G

itself.

Theorem 21.

Every group of prime order is simple.

Proof.

By Lagrange's theorem, if

∣ G ∣ = p

(prime), the only possible subgroup orders are

1

and

p

. Hence the only subgroups are

{e}

and

G

, so

G

has no proper normal subgroups. □

Theorem 22.

For

n \geq 5

, the alternating group

A_{n}

is simple.

Remark 23.

The proof that

A_{n}

is simple for

n \geq 5

requires a substantial argument, which we omit here. The group

A_{5}

is the smallest non-abelian simple group, with

∣ A_{5} ∣ = 60

Remark 24.

9 Analyzing Structure via Quotient Groups

Theorem 25 (Correspondence theorem (Fourth isomorphism theorem)).

Let

N ⊴ G

. There is an inclusion-preserving bijection between the subgroups of

G

that contain

N

and the subgroups of

G / N

H \mapsto H / N, \overset{ˉ}{H} \mapsto π^{- 1} (\overset{ˉ}{H}),

where

π : G \to G / N

is the canonical projection. Moreover,

H ⊴ G

if and only if

H / N ⊴ G / N

Proof.

N \leq H \leq G

, then

N ⊴ H

(since

N ⊴ G

and

N \leq H

), so

H / N

is a well-defined group, and

H / N \leq G / N

Conversely, if

\overset{ˉ}{H} \leq G / N

, then

π^{- 1} (\overset{ˉ}{H}) = {g \in G ∣ g N \in \overset{ˉ}{H}}

is a subgroup of

G

containing

N

One verifies that these two correspondences are mutual inverses and preserve both inclusion and normality. □

10 Summary and Next Steps

In this chapter we have covered:

The definition of normal subgroups and their equivalent characterizations (closure under conjugation).
Examples: abelian groups, subgroups of index $2$ , kernels, $A_{n}$ , $S L_{n}$ .
Construction of the quotient group $G / N$ and the proof of well-definedness.
The canonical projection $π : G \to G / N$ and the correspondence "normal subgroups $=$ kernels."
Simple groups: definition and examples.
The correspondence theorem.

Group Theory Algebra Textbook Normal Subgroups Quotient Groups

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Normal Subgroups and Quotient Groups

1 Motivation

2 Definition of Normal Subgroups

3 Equivalent Conditions for Normality

4 Examples of Normal Subgroups

5 Construction of the Quotient Group

6 The Canonical Projection

7 Homomorphisms and Normal Subgroups

8 Simple Groups

9 Analyzing Structure via Quotient Groups

10 Summary and Next Steps

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Normal Subgroups and Quotient Groups

1 Motivation

2 Definition of Normal Subgroups

3 Equivalent Conditions for Normality

4 Examples of Normal Subgroups

5 Construction of the Quotient Group

6 The Canonical Projection

7 Homomorphisms and Normal Subgroups

8 Simple Groups

9 Analyzing Structure via Quotient Groups

10 Summary and Next Steps

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