Composition Series and the Jordan--Hölder Theorem

We introduce normal series and composition series, establishing the framework for decomposing a group into simple factors. The Jordan--Hölder theorem proves the uniqueness of the composition factors, and we develop the theory of solvable groups with its connections to the derived series.

Folio Official

March 1, 2026

1 Normal Series

Definition 1 (Normal series).

A normal series (or subnormal series) of a group

G

is a chain of subgroups

{e} = G_{0} ⊴ G_{1} ⊴ G_{2} ⊴ \dots ⊴ G_{n} = G .

The integer

n

is the length of the series, and the quotient groups

G_{i + 1} / G_{i}

are called the factors of the series.

Definition 2 (Refinement).

A refinement of a normal series is a normal series obtained by inserting additional subgroups between the existing terms.

Example 3.

The group

G = Z /12 Z

has the normal series

{0} ⊴ ⟨ 6 ⟩ ⊴ ⟨ 3 ⟩ ⊴ ⟨ 1 ⟩ = Z /12 Z .

The factors are

⟨ 6 ⟩ / {0} ≅ Z /2 Z

⟨ 3 ⟩ / ⟨ 6 ⟩ ≅ Z /2 Z

, and

Z /12 Z / ⟨ 3 ⟩ ≅ Z /3 Z

2 Simple Groups

Definition 4 (Simple group).

A group

G

of order at least

2

is simple if its only normal subgroups are

{e}

and

G

itself.

Simple groups are the groups that cannot be decomposed further — they play the role of prime numbers in the world of group theory.

Example 5.

Every group $Z / p Z$ of prime order $p$ is simple.
The alternating group $A_{n}$ is simple for $n \geq 5$ . The simplicity of $A_{5}$ is the key to proving that the general quintic polynomial has no algebraic solution (via Galois theory).
$Z /4 Z$ is not simple, since ${0, 2}$ is a proper normal subgroup.

3 Composition Series

Definition 6 (Composition series).

A normal series

{e} = G_{0} ⊴ G_{1} ⊴ G_{2} ⊴ \dots ⊴ G_{n} = G

is a composition series if every factor

G_{i + 1} / G_{i}

is a simple group. The factors

G_{i + 1} / G_{i}

are then called the composition factors of

G

A composition series is a normal series that cannot be refined further.

Proposition 7.

A normal series is a composition series if and only if it admits no proper refinement.

Proof.

G_{i + 1} / G_{i}

is not simple, then it has a non-trivial proper normal subgroup, which corresponds to a subgroup

N

with

G_{i} ⪇ N ⪇ G_{i + 1}

and

N ⊴ G_{i + 1}

; inserting

N

gives a proper refinement. Conversely, if each factor is simple, no subgroup can be inserted between consecutive terms. □

Theorem 8.

Every finite group has a composition series.

Proof.

By induction on

∣ G ∣

. If

G

is simple, then

{e} ⊴ G

is a composition series. If

G

is not simple, there exists a non-trivial proper normal subgroup

N

. Since

∣ N ∣ < ∣ G ∣

and

∣ G / N ∣ < ∣ G ∣

, the induction hypothesis applies to both, and we can concatenate their composition series to obtain one for

G

. □

Example 9 (Composition series of

S_{3}

{e} ⊴ A_{3} ⊴ S_{3} .

The composition factors are

A_{3} / {e} ≅ Z /3 Z

and

S_{3} / A_{3} ≅ Z /2 Z

Example 10 (Composition series of

S_{4}

A first attempt at a normal series is

{e} ⊴ V_{4} ⊴ A_{4} ⊴ S_{4}

where

V_{4} = {e, (12) (34), (13) (24), (14) (23)}

is the Klein four-group. But

V_{4} / {e} ≅ V_{4}

is not simple, so this is not a composition series. Refining further:

{e} ⊴ ⟨(12) (34)⟩ ⊴ V_{4} ⊴ A_{4} ⊴ S_{4} .

The composition factors are

Z /2 Z

Z /2 Z

Z /3 Z

, and

Z /2 Z

4 The Jordan–Hölder Theorem

Theorem 11 (Jordan–H\"older Theorem).

Let

G

be a finite group with two composition series

{e} = G_{0} ⊴ G_{1} ⊴ \dots ⊴ G_{m} = G, {e} = H_{0} ⊴ H_{1} ⊴ \dots ⊴ H_{n} = G .

Then

m = n

, and the composition factors from the two series agree up to permutation and isomorphism:

{G_{1} / G_{0}, G_{2} / G_{1}, \dots, G_{m} / G_{m - 1}} = {H_{1} / H_{0}, H_{2} / H_{1}, \dots, H_{n} / H_{n - 1}}

as multisets of isomorphism classes.

This theorem is the group-theoretic analogue of the fundamental theorem of arithmetic (uniqueness of prime factorization of integers).

Proof.

We proceed by induction on

m + n

Case 1:

G_{m - 1} = H_{n - 1}

. Then the two composition series share the same penultimate term, and the result follows by applying the induction hypothesis to the composition series of

G_{m - 1}

Case 2:

G_{m - 1} \neq = H_{n - 1}

. Both

G_{m - 1}

and

H_{n - 1}

are maximal normal subgroups of

G

(since

G / G_{m - 1}

and

G / H_{n - 1}

are simple). Set

K = G_{m - 1} \cap H_{n - 1}

Since

G_{m - 1}

and

H_{n - 1}

are both maximal and distinct,

G_{m - 1} H_{n - 1} = G

. The second isomorphism theorem gives

G / G_{m - 1} G / H_{n - 1} ≅ H_{n - 1} / (G_{m - 1} \cap H_{n - 1}) = H_{n - 1} / K, ≅ G_{m - 1} / (G_{m - 1} \cap H_{n - 1}) = G_{m - 1} / K .

Take a composition series for

K

and extend it through

G_{m - 1}

and through

H_{n - 1}

to form two new composition series of

G

. Applying the induction hypothesis to

G_{m - 1}

and

H_{n - 1}

shows that the multisets of composition factors agree. □

5 Solvable Groups

Definition 12 (Solvable group).

A group

G

is solvable if it possesses a normal series

{e} = G_{0} ⊴ G_{1} ⊴ \dots ⊴ G_{n} = G

whose factors

G_{i + 1} / G_{i}

are all abelian.

Proposition 13.

For a finite group

G

, the following are equivalent:

$G$ is solvable.
Every composition factor of $G$ is a cyclic group of prime order.

Proof.

(2) \Rightarrow (1)

: Every cyclic group of prime order is abelian.

(1) \Rightarrow (2)

: The composition factors of an abelian group are simple abelian groups, hence cyclic of prime order (by the structure theorem for finite abelian groups). □

Example 14.

$S_{3}$ is solvable: ${e} ⊴ A_{3} ⊴ S_{3}$ with factors $Z /3 Z$ and $Z /2 Z$ , both abelian.
$S_{4}$ is solvable: ${e} ⊴ V_{4} ⊴ A_{4} ⊴ S_{4}$ with factors $V_{4} ≅ (Z /2 Z)^{2}$ , $Z /3 Z$ , and $Z /2 Z$ , all abelian.
$S_{n}$ for $n \geq 5$ is not solvable: $A_{n}$ is a simple non-abelian group, so it appears as a composition factor that is not cyclic of prime order.

6 The Derived Series

Definition 15 (Commutator subgroup).

The commutator subgroup (or derived subgroup)

G^{'} = [G, G]

of a group

G

is defined as

G^{'} = ⟨ g, h ∣ g, h \in G ⟩, where g, h = g^{- 1} h^{- 1} g h .

Proposition 16.

G^{'} ⊴ G

, and

G / G^{'}

is abelian. Moreover,

G / G^{'}

is the largest abelian quotient of

G

: a quotient

G / N

is abelian if and only if

G^{'} \leq N

Proof.

Every automorphism

φ

G

maps

[g, h]

[φ (g), φ (h)]

, so

G^{'}

is a characteristic subgroup and in particular normal. To see that

G / G^{'}

is abelian, note that

(g G^{'}) (h G^{'}) = g h G^{'} = h g [g^{- 1}, h^{- 1}] G^{'} = h g G^{'}

since

[g^{- 1}, h^{- 1}] \in G^{'}

G / N

is abelian, then

g h N = h g N

for all

g, h

, so

g^{- 1} h^{- 1} g h \in N

for all

g, h

. Thus

G^{'} \leq N

. □

Definition 17 (Derived series).

The derived series of

G

is defined inductively by

G^{(0)} = G, G^{(1)} = G^{'} = G, G, G^{(k + 1)} = G^{(k)}, G^{(k)} .

Theorem 18.

A finite group

G

is solvable if and only if

G^{(n)} = {e}

for some

n

Proof.

(\Rightarrow)

Let

{e} = G_{0} ⊴ G_{1} ⊴ \dots ⊴ G_{r} = G

be a series with abelian factors. Since

G_{i + 1} / G_{i}

is abelian, we have

G_{i + 1}^{'} \leq G_{i}

. Therefore

G^{(1)} = G^{'} \leq G_{r - 1}

G^{(2)} \leq G_{r - 2}

, and so on, giving

G^{(r)} \leq G_{0} = {e}

(\Leftarrow)

G^{(n)} = {e}

, then

G = G^{(0)} ⊵ G^{(1)} ⊵ \dots ⊵ G^{(n)} = {e}

is a normal series with abelian factors (since

G^{(k)} / G^{(k + 1)}

is always abelian). □

7 Properties of Solvable Groups

Theorem 19.

If $G$ is solvable and $H \leq G$ , then $H$ is solvable.
If $G$ is solvable and $N ⊴ G$ , then $G / N$ is solvable.
If $N ⊴ G$ and both $N$ and $G / N$ are solvable, then $G$ is solvable.

Proof.

(1) One shows inductively that

H^{(k)} \leq G^{(k)}

. If

G^{(n)} = {e}

, then

H^{(n)} = {e}

(2) The natural surjection

π : G \to G / N

satisfies

π (G^{(k)}) = (G / N)^{(k)}

, as is proved by induction. Hence

G^{(n)} = {e}

implies

(G / N)^{(n)} = {e}

(3) Since

G / N

is solvable,

(G / N)^{(m)} = {e}

for some

m

, which means

G^{(m)} \leq N

. Since

N

is solvable,

N^{(k)} = {e}

for some

k

. Then

G^{(m + k)} = (G^{(m)})^{(k)} \leq N^{(k)} = {e}

. □

Theorem 20 (Burnside's theorem).

Every finite group of order

p^{a} q^{b}

, where

p

and

q

are primes, is solvable.

The proof of this theorem requires representation theory (specifically, character theory) and is beyond our present scope. A purely group-theoretic proof was not known for a long time. A far-reaching generalization is the Feit–Thompson theorem (the odd-order theorem): every finite group of odd order is solvable.

8 Summary

A composition series decomposes a group into simple factors.
The Jordan–Hölder theorem asserts that the multiset of composition factors is an invariant of the group — it is the "prime factorization" of the group.
A group is solvable if all its composition factors are cyclic of prime order. Solvability can be tested via the derived series.
$S_{n}$ is solvable for $n \leq 4$ but not for $n \geq 5$ (because $A_{n}$ is a non-abelian simple group).
The structure of a group separates into two questions: which simple groups appear as composition factors, and how they are assembled (the extension problem).

Group Theory Algebra Textbook Composition Series Jordan-Hölder Theorem Simple Groups

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Composition Series and the Jordan--Hölder Theorem

1 Normal Series

2 Simple Groups

3 Composition Series

4 The Jordan–Hölder Theorem

5 Solvable Groups

6 The Derived Series

7 Properties of Solvable Groups

8 Summary

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