Group Actions and the Class Equation

We introduce group actions and develop the orbit-stabilizer theorem, Burnside's lemma, and the class equation. Applications include the proof that the center of a p-group is nontrivial, the classification of groups of order p^2, the fixed-point theorem for p-groups, and Cayley's theorem.

Folio Official

March 1, 2026

1 Definition of a Group Action

Definition 1 (Group action).

Let

G

be a group and

X

a set. A map

G \times X \to X

, written

(g, x) \mapsto g \cdot x

, is called a (left) action of

G

X

if the following two conditions hold:

$e \cdot x = x$ for all $x \in X$ .
$(g h) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$ .

When

G

acts on

X

, we call

X

G

-set.

2 Fundamental Examples

Example 2 (Trivial action).

Setting

g \cdot x = x

for all

g \in G

and

x \in X

defines a group action, called the trivial action.

Example 3 (Left regular representation).

The group

G

acts on itself by left multiplication:

g \cdot x = gx

. Axiom (1):

e \cdot x = e x = x

. Axiom (2):

(g h) \cdot x = (g h) x = g (h x) = g \cdot (h \cdot x)

Example 4 (Conjugation action).

G

acts on itself by conjugation:

g \cdot x = gx g^{- 1}

. Axiom (1):

e \cdot x = e x e^{- 1} = x

. Axiom (2):

(g h) \cdot x = (g h) x (g h)^{- 1} = g (h x h^{- 1}) g^{- 1} = g \cdot (h \cdot x)

Example 5 (Action on subgroups by conjugation).

G

acts on the set of subgroups of

G

g \cdot H = g H g^{- 1}

Example 6 (Action on cosets).

For

H \leq G

, the group

G

acts on the set of left cosets

G / H = {g H ∣ g \in G}

g \cdot (a H) = (g a) H

Example 7 (The natural action of

S_{n}

S_{n}

acts on

{1, 2, \dots, n}

σ \cdot i = σ (i)

3 Orbits and Stabilizers

Definition 8 (Orbit).

Let

G

act on

X

. The orbit of

x \in X

Orb (x) = G \cdot x = {g \cdot x ∣ g \in G} .

Definition 9 (Stabilizer).

The stabilizer (or isotropy subgroup) of

x \in X

Stab (x) = G_{x} = {g \in G ∣ g \cdot x = x} .

Theorem 10.

Stab (x)

is a subgroup of

G

Proof.

Since

e \cdot x = x

, we have

e \in Stab (x)

. If

g, h \in Stab (x)

, then

(g h^{- 1}) \cdot x = g \cdot (h^{- 1} \cdot x) = g \cdot x = x

, where the equality

h^{- 1} \cdot x = x

follows from

h \cdot (h^{- 1} \cdot x) = (h h^{- 1}) \cdot x = e \cdot x = x

. Hence

g h^{- 1} \in Stab (x)

. □

Theorem 11 (Orbits partition

X

The relation

x \sim y ⟺ y \in Orb (x)

is an equivalence relation on

X

, and the orbits form a partition of

X

X = i ⨆ Orb (x_{i}) (choosing one representative from each orbit).

Proof.

Reflexivity:

x = e \cdot x \in Orb (x)

. Symmetry: if

y = g \cdot x

, then

x = g^{- 1} \cdot y \in Orb (y)

. Transitivity: if

y = g \cdot x

and

z = h \cdot y

, then

z = (h g) \cdot x \in Orb (x)

. □

4 The Orbit–Stabilizer Theorem

Theorem 12 (Orbit–stabilizer theorem).

Let

G

act on

X

, and let

x \in X

. Then

∣ Orb (x) ∣ = [G : Stab (x)] .

In particular, if

G

is finite,

∣ Orb (x) ∣ = ∣ G ∣/∣ Stab (x) ∣

Proof.

We construct a bijection

f : G / Stab (x) \to Orb (x)

by setting

f (g Stab (x)) = g \cdot x

Well-definedness. If

g Stab (x) = h Stab (x)

, then

h^{- 1} g \in Stab (x)

, so

(h^{- 1} g) \cdot x = x

, whence

g \cdot x = h \cdot x

Injectivity. If

g \cdot x = h \cdot x

, then

(h^{- 1} g) \cdot x = x

, so

h^{- 1} g \in Stab (x)

, giving

g Stab (x) = h Stab (x)

Surjectivity. Every

g \cdot x \in Orb (x)

is the image of

g Stab (x)

. □

Example 13.

S_{3}

acts naturally on

{1, 2, 3}

. For

x = 1

Orb (1) = {1, 2, 3}

(the full set), and

Stab (1) = {e, (23)}

. Indeed,

∣ Orb (1) ∣ = 3 = 6/2 = ∣ S_{3} ∣/∣ Stab (1) ∣

5 Counting Orbits: Burnside's Lemma

Definition 14 (Fixed-point set).

For

g \in G

, the fixed-point set of

g

Fix (g) = X^{g} = {x \in X ∣ g \cdot x = x}

Theorem 15 (Burnside's lemma).

Let

G

be a finite group acting on a finite set

X

. The number of orbits

∣ X / G ∣

is given by

∣ X / G ∣ = \frac{1}{∣ G ∣} g \in G \sum ∣ Fix (g) ∣.

Proof.

Count the elements of the set

S = {(g, x) \in G \times X ∣ g \cdot x = x}

in two ways. Summing over

g

gives

\sum_{g \in G} ∣ Fix (g) ∣

. Summing over

x

gives

\sum_{x \in X} ∣ Stab (x) ∣

By the orbit–stabilizer theorem,

∣ Stab (x) ∣ = ∣ G ∣/∣ Orb (x) ∣

, so

x \in X \sum ∣ Stab (x) ∣ = x \in X \sum \frac{∣ G ∣}{∣ Orb ( x ) ∣} = ∣ G ∣ x \in X \sum \frac{1}{∣ Orb ( x ) ∣} .

Elements in the same orbit contribute the same value

1/∣ Orb (x) ∣

, and each orbit of size

k

contributes

k \cdot 1/ k = 1

. So the right-hand side equals

∣ G ∣ \cdot ∣ X / G ∣

. □

6 Conjugacy Classes and the Class Equation

Applying the orbit–stabilizer theorem to the conjugation action $g \cdot x = gx g^{- 1}$ yields powerful structural results.

Definition 16 (Conjugacy class and centralizer).

The orbit of an element

a

under the conjugation action is called its conjugacy class:

C (a) = {g a g^{- 1} ∣ g \in G} .

The stabilizer of

a

under conjugation is called the centralizer of

a

C_{G} (a) = {g \in G ∣ g a g^{- 1} = a} = {g \in G ∣ g a = a g} .

Definition 17 (Center of a group).

The center of

G

Z (G) = {z \in G ∣ z g = g z for all g \in G} = a \in G ⋂ C_{G} (a) .

One readily verifies that

Z (G) ⊴ G

Remark 18.

An element

a

lies in

Z (G)

if and only if its conjugacy class is the singleton

{a}

Theorem 19 (The class equation).

Let

G

be a finite group. Decomposing

G

into conjugacy classes gives

∣ G ∣ = ∣ Z (G) ∣ + i \sum [G : C_{G} (a_{i})],

where the sum runs over one representative

a_{i}

from each conjugacy class of size greater than

1

(i.e.

a_{i} \in / Z (G)

). Each summand

[G : C_{G} (a_{i})] \geq 2

and divides

∣ G ∣

Proof.

G

decomposes as a disjoint union of conjugacy classes. The size of the conjugacy class of

a

∣ C (a) ∣ = [G : C_{G} (a)]

(by the orbit–stabilizer theorem). A conjugacy class has size

1

precisely when

a \in Z (G)

. Collecting the size-

1

classes and the others:

∣ G ∣ = ∣ Z (G) ∣ + i \sum ∣ C (a_{i}) ∣ = ∣ Z (G) ∣ + i \sum [G : C_{G} (a_{i})] .

□

7 Properties of $p$ -Groups

Definition 20 (

p

-group).

For a prime

p

, a finite group

G

with

∣ G ∣ = p^{n}

(

n \geq 1

) is called a

p

-group.

Theorem 21 (The center of a

p

-group is nontrivial).

Let

G

be a

p

-group with

∣ G ∣ = p^{n}

n \geq 1

. Then

Z (G) \neq = {e}

Proof.

In the class equation

∣ G ∣ = ∣ Z (G) ∣ + \sum_{i} [G : C_{G} (a_{i})]

, each term

[G : C_{G} (a_{i})]

divides

∣ G ∣ = p^{n}

(by Lagrange's theorem) and is at least

2

, so

p ∣ [G : C_{G} (a_{i})]

. Since

p ∣ ∣ G ∣

∣ Z (G) ∣ = ∣ G ∣ - i \sum [G : C_{G} (a_{i})] \equiv 0 (mod p) .

Since

e \in Z (G)

, we have

∣ Z (G) ∣ \geq 1

, and combined with

p ∣ ∣ Z (G) ∣

, we get

∣ Z (G) ∣ \geq p

. In particular,

Z (G) \neq = {e}

. □

Theorem 22 (Groups of order

p^{2}

are abelian).

∣ G ∣ = p^{2}

, then

G

is abelian.

Proof.

Since

Z (G) \neq = {e}

, we have

∣ Z (G) ∣ = p

p^{2}

. If

∣ Z (G) ∣ = p^{2}

, then

Z (G) = G

and

G

is abelian.

Suppose

∣ Z (G) ∣ = p

. Then

∣ G / Z (G) ∣ = p

, so

G / Z (G)

is cyclic; say

G / Z (G) = ⟨ g Z (G)⟩

. Every element of

G

can be written as

g^{i} z

for some integer

i

and some

z \in Z (G)

. For any two elements

a = g^{i} z_{1}

and

b = g^{j} z_{2}

(with

z_{1}, z_{2} \in Z (G)

ab = g^{i} z_{1} g^{j} z_{2} = g^{i + j} z_{1} z_{2} = g^{j} z_{2} g^{i} z_{1} = ba,

since elements of

Z (G)

commute with everything. This shows

G

is abelian, so

Z (G) = G

, contradicting

∣ Z (G) ∣ = p

. □

Theorem 23 (Fixed-point theorem for

p

-groups).

Let

G

be a

p

-group acting on a finite set

X

. Let

X^{G} = {x \in X ∣ g \cdot x = x for all g \in G}

denote the set of fixed points. Then

∣ X^{G} ∣ \equiv ∣ X ∣ (mod p) .

Proof.

In the orbit decomposition

X = X^{G} ⊔ ⨆_{i} Orb (x_{i})

(where

∣ Orb (x_{i}) ∣ \geq 2

), each orbit size

∣ Orb (x_{i}) ∣ = [G : Stab (x_{i})]

divides

∣ G ∣ = p^{n}

and is at least

2

, hence is divisible by

p

. Therefore:

∣ X ∣ = ∣ X^{G} ∣ + i \sum ∣ Orb (x_{i}) ∣ \equiv ∣ X^{G} ∣ (mod p) .

□

8 Cayley's Theorem

Theorem 24 (Cayley's theorem).

Every group

G

is isomorphic to a subgroup of a symmetric group. If

G

is finite with

∣ G ∣ = n

, then

G

is isomorphic to a subgroup of

S_{n}

Proof.

Let

G

act on itself by left multiplication:

g \cdot x = gx

. For each

g \in G

, define

σ_{g} : G \to G

σ_{g} (x) = gx

. This is a bijection (with inverse

σ_{g^{- 1}}

), so

σ_{g} \in S_{G}

Define

φ : G \to S_{G}

φ (g) = σ_{g}

. This is a homomorphism:

φ (g h) (x) = σ_{g h} (x) = (g h) x = g (h x) = σ_{g} (σ_{h} (x)) = (φ (g) \circ φ (h)) (x) .

Furthermore,

ker φ = {g \in G ∣ gx = x for all x \in G} = {e}

, so

φ

is injective. By the first isomorphism theorem,

G ≅ Im φ \leq S_{G}

. □

9 Summary and Next Steps

In this chapter we have covered:

The definition of a group action and its fundamental examples.
Orbits, stabilizers, and the orbit–stabilizer theorem.
Burnside's lemma for counting orbits.
Conjugacy classes, centralizers, the center, and the class equation.
The center of a $p$ -group is nontrivial; groups of order $p^{2}$ are abelian.
The fixed-point theorem for $p$ -groups.
Cayley's theorem.

With the class equation and the fixed-point theorem for $p$ -groups in hand, we are now ready to prove one of the pinnacles of finite group theory: the Sylow theorems (existence, conjugacy, and the counting theorem for Sylow $p$ -subgroups).

Group Theory Algebra Textbook Group Actions Class Equation Burnside's Lemma

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1 フォロワー·105 記事

Group Actions and the Class Equation

1 Definition of a Group Action

2 Fundamental Examples

3 Orbits and Stabilizers

4 The Orbit–Stabilizer Theorem

5 Counting Orbits: Burnside's Lemma

6 Conjugacy Classes and the Class Equation

7 Properties of $p$ -Groups

8 Cayley's Theorem

9 Summary and Next Steps

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1 Definition of a Group Action

2 Fundamental Examples

3 Orbits and Stabilizers

4 The Orbit–Stabilizer Theorem

5 Counting Orbits: Burnside's Lemma

6 Conjugacy Classes and the Class Equation

7 Properties ofp-Groups

8 Cayley's Theorem

9 Summary and Next Steps

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