Actions, Orbits, and Cayley's Theorem

A group is defined by axioms, but much of its structure is visible only in how it acts. The same abstract group can permute the vertices of a polygon, rearrange the roots of a polynomial, or shuffle its own elements by multiplication. Each is a group action, and orbits, stabilizers, and the counting relation between them extract structure from a group.¹

Actions and permutation representations

A group homomorphism maps one group into another. An action instead sends a group into the symmetries of a bare set.

Fix $g \in G$ and let $σ_{g} : A \to A$ be the map $a \mapsto g \cdot a$ . The two axioms make $σ_{g}$ a bijection: $σ_{g^{- 1}}$ is its two-sided inverse, since $σ_{g^{- 1}} \circ σ_{g} = σ_{g^{- 1} g} = σ_{1} = id$ . So every element acts as a permutation of $A$ , and the assignment $g \mapsto σ_{g}$ is a homomorphism $φ : G \to S_{A}$ into the symmetric group on $A$ , because $σ_{g_{1} g_{2}} = σ_{g_{1}} \circ σ_{g_{2}}$ restates the first axiom. This homomorphism is the permutation representation afforded by the action.

The correspondence runs both ways.²

An action is thus a homomorphism into a symmetric group, written in operator notation. Three subsets record what the action leaves fixed.

Kernel: $ker φ = {g \in G : g \cdot a = a for all a \in A}$ , the elements acting trivially on everything. It is a normal subgroup, being the kernel of $φ$ .
Stabilizer of a point $a$ : $G_{a} = {g \in G : g \cdot a = a}$ , the elements fixing that one point. Each $G_{a}$ is a subgroup of $G$ , and $ker φ = ⋂_{a \in A} G_{a}$ .
Faithful action: one whose kernel is trivial, so distinct elements induce distinct permutations and $φ$ is injective. A faithful action embeds $G$ as a subgroup of $S_{A}$ .

When the action is not faithful, $G / ker φ$ acts faithfully on $A$ ; the kernel is the only obstruction to embedding $G$ in $S_{A}$ .

For example, let $G = D_{8}$ act on the four vertices of a square labelled $1, 2, 3, 4$ clockwise. The clockwise rotation $r$ by $π /2$ induces $(1234)$ , and the reflection $s$ through vertices $1$ and $3$ induces $(24)$ . This action is faithful (only the identity symmetry fixes all four vertices), and the stabilizer of vertex $1$ is $⟨ s ⟩$ , the order- $2$ subgroup generated by the reflection through it.

Orbits and the equivalence they define

Points reachable from one another by the group form one class.

Reflexivity is $a = 1 \cdot a$ ; symmetry follows by acting with $g^{- 1}$ ; and transitivity chains two group elements into their product. The orbit of $a$ is $O_{a} = {g \cdot a : g \in G}$ . The action is transitive when there is a single orbit, i.e. any point can be moved to any other.

The size of an orbit is the index of a stabilizer.

The map is well defined and injective because $g \cdot a = h \cdot a$ holds exactly when $h^{- 1} g$ fixes $a$ , i.e. when $g G_{a} = h G_{a}$ ; it is surjective by the definition of the orbit.³ The finite form is Lagrange's theorem rewritten: a large orbit forces a small stabilizer and vice versa.

Orbit-stabilizer as a bijection: the four left cosets of the stabilizer

G_{a}

each map to one point of the orbit, so

∣ G ∣ = ∣ O_{a} ∣ \cdot ∣ G_{a} ∣

Cycle decomposition is an orbit decomposition

The abstract statement recovers a concrete fact from permutation groups. Let $σ \in S_{n}$ and set $G = ⟨ σ ⟩$ acting on $A = {1, \dots, n}$ . The orbits partition $A$ , and on an orbit of size $d$ the generator $σ$ sends

x, σ (x), σ^{2} (x), \dots, σ^{d - 1} (x), σ^{d} (x) = x,

so $σ$ acts as a single $d$ -cycle there. The orbits of $⟨ σ ⟩$ are the cycles of $σ$ . This proves that every permutation has a cycle decomposition, unique up to reordering the cycles and rotating within each. For instance, $⟨(12) (345)⟩$ acting on ${1, \dots, 5}$ has orbits ${1, 2}$ and ${3, 4, 5}$ .

The cyclic group generated by

(12) (345)

splits

{1, \dots, 5}

into two orbits; the generator runs each orbit as one cycle.

Computing an orbit

Orbit-stabilizer is also an algorithm. To list an orbit, grow it from the seed point by applying generators until nothing new appears — a breadth-first search on the set $A$ .

Algorithm:

\textsc{Orbit}(a, S)

— the orbit of

a

under the group generated by

S

1
initialize $O \gets \{a\}$ and a queue $Q \gets [a]$
2
while $Q$ is nonempty do
3
dequeue $x$ from $Q$
4
for each generator $s \in S$ do
5
$y \gets s \cdot x$
6
if $y \notin O$ then
7
add $y$ to $O$ and enqueue $y$
8
return $O$

Since every element of $G$ is a word in the generators $S$ , the search reaches every point $g \cdot a$ , and $∣ O ∣ = [G : G_{a}]$ on termination. The same traversal, recording which generator first reached each point, produces coset representatives for the stabilizer.

The action on vertices, and transitivity

For a running geometric example, take $D_{8}$ acting on the four vertices of a square. The action is transitive: any vertex reaches any other under some symmetry, so there is a single orbit of size $4$ . Orbit-stabilizer then reads $∣ D_{8} ∣ = 4 \cdot ∣ G_{1} ∣$ , forcing $∣ G_{1} ∣ = 2$ — the stabilizer of a vertex is the order- $2$ reflection through it.

D_{8}

acts transitively on the square's four vertices (one orbit of size

4

); the stabilizer of a vertex is the reflection through it, of order

2

Transitivity is not inherited by subgroups: the subgroup $⟨(12), (34)⟩ \leq S_{4}$ acts on ${1, 2, 3, 4}$ with two orbits, ${1, 2}$ and ${3, 4}$ , since no element sends $2$ to $3$ . Transitivity is a property of the whole group's reach.

Cayley's theorem: every group is a permutation group

The most consequential action is a group acting on itself. Let $G$ act on the set $A = G$ by left multiplication, $g \cdot a = g a$ . The axioms hold because $g_{1} (g_{2} a) = (g_{1} g_{2}) a$ and $1 \cdot a = a$ . This action is transitive (given $a, b$ , the element $b a^{- 1}$ sends $a$ to $b$ ) and faithful (if $g a = a$ for all $a$ then $g = 1$ ), and the stabilizer of every point is trivial.

The permutation representation afforded by left multiplication is the left regular representation $G \to S_{G}$ . Because the action is faithful, this homomorphism is injective.⁴

The proof is immediate: left multiplication gives an injective homomorphism $G \to S_{G}$ , and $G$ is isomorphic to its image, a subgroup of $S_{G} ≅ S_{n}$ . The historical import is that the modern axiomatic group and the nineteenth-century notion of a group of substitutions coincide.

The left regular representation of the Klein four-group: each element, acting by left multiplication, becomes a fixed-point-free permutation in

S_{4}

Cayley's theorem is more a statement of principle than a computational tool: studying a group of order $n$ inside $S_{n}$ is rarely efficient, but it certifies that permutations lose no generality.

Actions on cosets and a normality test

Left multiplication generalizes from $G$ itself to the coset space of any subgroup. Let $H \leq G$ and let $G$ act on the set $A$ of left cosets of $H$ by $g \cdot a H = (g a) H$ .⁵

Transitivity holds because $b a^{- 1}$ carries $a H$ to $b H$ . The stabilizer of $1 H$ is ${g : g H = H} = H$ . For the kernel, $g$ fixes every coset $x H$ exactly when $x^{- 1} g x \in H$ for all $x$ , i.e. $g \in ⋂_{x} x H x^{- 1}$ ; any normal subgroup inside $H$ lands in this intersection, so it is the largest one. Taking $H = 1$ recovers Cayley's theorem.

A subgroup of small enough index is forced to be normal.

Let $K = ker π_{H}$ . Then $G / K$ embeds in $S_{p}$ , so $[G : K]$ divides $p!$ . Writing $[H : K] = k$ gives $[G : K] = p k$ , and $p k ∣ p!$ forces $k ∣ (p - 1)!$ . Every prime divisor of $(p - 1)!$ is less than $p$ , while every prime divisor of $k$ (a divisor of $∣ G ∣$ ) is at least $p$ ; the only escape is $k = 1$ , so $H = K ⊴ G$ .⁶ The classical index- $2$ result is the case $p = 2$ .

Action of $G$	Set acted on	Transitive?	Stabilizer of a point	Kernel
Left multiplication	$G$	yes	$1$	$1$ (faithful)
Left mult. on cosets	cosets of $H$	yes	$H$	$⋂_{x} x H x^{- 1}$
Natural action of $S_{n}$	${1, \dots, n}$	yes	$S_{n - 1}$	$1$
Conjugation	$G$	no	$C_{G} (a)$	$Z (G)$

The last row is the conjugation action: $g \cdot a = g a g^{- 1}$ replaces multiplication, the orbits become conjugacy classes, and orbit-stabilizer applied to it produces the class equation.

Dummit & Foote, §4.1 — Group Actions and Permutation Representations: the axioms of an action, the associated homomorphism into $S_{A}$ , and the kernel/stabilizer/faithful terminology. ↩
Dummit & Foote, §4.1, Proposition 1: the bijection between actions of $G$ on $A$ and homomorphisms $G \to S_{A}$ . ↩
Dummit & Foote, §4.1, Proposition 2: the orbit relation is an equivalence relation and $∣ O_{a} ∣ = [G : G_{a}]$ , together with the cycle-decomposition application. ↩ ↩²
Dummit & Foote, §4.2, Corollary 4: Cayley's theorem via the left regular representation. ↩
Dummit & Foote, §4.2, Theorem 3: the action on left cosets of $H$ , its transitivity, stabilizer, and kernel as the largest normal subgroup inside $H$ . ↩
Dummit & Foote, §4.2, Corollary 5: a subgroup of index equal to the smallest prime divisor of $∣ G ∣$ is normal. ↩