---
title: Actions, Orbits, and Cayley's Theorem
module: Group Actions and Sylow Theory
moduleNumber: 4
lessonNumber: 1
order: 401
summary: >
  A group action turns abstract elements into permutations of a set. The action
  splits the set into orbits, and the orbit-stabilizer theorem ties each orbit's
  size to the index of a stabilizer. Applied to a group acting on itself by left
  multiplication, this gives Cayley's theorem: every group is a group of
  permutations.
topics: [Group Actions and Sylow Theory]
sources:
  - book: Dummit & Foote
    ref: "Ch. 4 Group Actions; §4.1 Group Actions and Permutation Representations, §4.2 Left Multiplication and Cayley's Theorem"
  - book: Judson
    ref: "Ch. 14 Group Actions; §14.1 Groups Acting on Sets"
draft: false
---

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.[^df-intro]

## Actions and permutation representations

A group [homomorphism](/abstract-algebra/groups-and-symmetry/homomorphisms-and-group-actions)
maps one group into another. An action instead sends a group into the symmetries
of a bare set.

> **Definition (Group action).** A (left) **action** of a group $G$ on a nonempty
> set $A$ is a map $G \times A \to A$, written $g \cdot a$, satisfying
>
> $$
> g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a
> \qquad\text{and}\qquad
> 1 \cdot a = a
> $$
>
> for all $g_1, g_2 \in G$ and all $a \in A$.

Fix $g \in G$ and let $\sigma_g : A \to A$ be the map $a \mapsto g \cdot a$. The
two axioms make $\sigma_g$ a bijection: $\sigma_{g^{-1}}$ is its two-sided inverse,
since $\sigma_{g^{-1}} \circ \sigma_g = \sigma_{g^{-1}g} = \sigma_1 = \mathrm{id}$.
So every element acts as a permutation of $A$, and the assignment $g \mapsto
\sigma_g$ is a homomorphism $\varphi : G \to S_A$ into the symmetric group on $A$,
because $\sigma_{g_1 g_2} = \sigma_{g_1} \circ \sigma_{g_2}$ restates the first
axiom. This homomorphism is the **permutation representation** afforded by the
action.

The correspondence runs both ways.[^df-prop1]

> **Proposition (Actions are homomorphisms into $S_A$).** For any group $G$ and
> nonempty set $A$ there is a bijection between the actions of $G$ on $A$ and the
> homomorphisms $\varphi : G \to S_A$. Given $\varphi$, the formula
> $g \cdot a = \varphi(g)(a)$ recovers an action with the same permutation
> representation.

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

- **Kernel**: $\ker\varphi = \{\, g \in G : g \cdot a = a \text{ for all } a \in
  A \,\}$, the elements acting trivially on everything. It is a normal subgroup,
  being the kernel of $\varphi$.
- **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](/abstract-algebra/subgroups-and-quotients/subgroups-and-substructures)
  of $G$, and $\ker\varphi = \bigcap_{a \in A} G_a$.
- **Faithful action**: one whose kernel is trivial, so distinct elements induce
  distinct permutations and $\varphi$ is injective. A faithful action embeds $G$
  as a subgroup of $S_A$.

When the action is not faithful, $G / \ker\varphi$ 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 $\pi/2$ induces $(1\,2\,3\,4)$, and the
reflection $s$ through vertices $1$ and $3$ induces $(2\,4)$. This action is
faithful (only the identity symmetry fixes all four vertices), and the stabilizer
of vertex $1$ is $\langle s \rangle$, 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.

> **Proposition (Orbits partition the set).** For a group $G$ acting on $A$, the
> relation $a \sim b \iff a = g \cdot b$ for some $g \in G$ is an equivalence
> relation. Its classes are called the **orbits** of the action.

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
$\mathcal{O}_a = \{\, g \cdot a : g \in G \,\}$. 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.

> **Theorem (Orbit-Stabilizer).** Let $G$ act on $A$ and let $a \in A$. The map
> $g\, G_a \mapsto g \cdot a$ is a bijection from the left cosets of $G_a$ in $G$
> to the orbit $\mathcal{O}_a$. Hence
>
> $$
> |\mathcal{O}_a| = [G : G_a].
> $$
>
> If $G$ is finite, $|G| = |\mathcal{O}_a| \cdot |G_a|$.

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.[^df-prop2] The finite form is
[Lagrange's theorem](/abstract-algebra/subgroups-and-quotients/cosets-lagrange-and-normal-subgroups)
rewritten: a large orbit forces a small stabilizer and vice versa.

$$
% caption: Orbit-stabilizer as a bijection: the four left cosets of the stabilizer
% $G_a$ each map to one point of the orbit, so $|G| = |\mathcal{O}_a|\cdot|G_a|$.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\foreach \i/\lab in {0/{G_a}, 1/{g_2 G_a}, 2/{g_3 G_a}, 3/{g_4 G_a}} {
  \draw[fill=acc!10, draw=acc] (0,\i*0.9) rectangle (2.6,\i*0.9+0.8);
  \node at (1.3,\i*0.9+0.4) {$\lab$};
}
\node[anchor=south, font=\footnotesize] at (1.3,3.75) {$G$ (by cosets of $G_a$)};
\foreach \i/\lab in {0/{a_1}, 1/{a_2}, 2/{a_3}, 3/{a_4}} {
  \fill[acc] (6,\i*0.9+0.4) circle (2.8pt);
  \node[anchor=west] at (6.2,\i*0.9+0.4) {$\lab$};
}
\node[anchor=south, font=\footnotesize] at (6.3,3.75) {orbit of $a$};
\foreach \i in {0,1,2,3} {
  \draw[->, acc, thick] (2.75,\i*0.9+0.4) -- (5.85,\i*0.9+0.4);
}
\end{tikzpicture}
$$

> **Worked example.** The rotation group of a cube has order $24$. Let $G$ be its
> group of rotational symmetries, acting on the six faces. The action is transitive
> — some rotation carries any face to any other — so there is one orbit of size
> $6$. The stabilizer of a fixed face is the set of rotations holding it in place:
> the identity together with the turns by $90^\circ, 180^\circ, 270^\circ$ about the
> axis through that face and its opposite, so $|G_{\text{face}}| = 4$.
> Orbit-stabilizer gives
>
> $$
> |G| = |\mathcal{O}| \cdot |G_{\text{face}}| = 6 \cdot 4 = 24.
> $$
>
> The count is independent of which set $G$ acts on: through the eight vertices it
> reads $8 \cdot 3 = 24$ (three rotations fix each vertex), and through the twelve
> edges $12 \cdot 2 = 24$.[^df-prop2]

### Cycle decomposition is an orbit decomposition

The abstract statement recovers a concrete fact from
[permutation groups](/abstract-algebra/groups-and-symmetry/dihedral-and-symmetric-groups).
Let $\sigma \in S_n$ and set $G = \langle \sigma \rangle$ acting on $A = \{1,
\dots, n\}$. The orbits partition $A$, and on an orbit of size $d$ the generator
$\sigma$ sends
$$
x,\ \sigma(x),\ \sigma^2(x),\ \dots,\ \sigma^{d-1}(x),\ \sigma^d(x) = x,
$$
so $\sigma$ acts as a single $d$-cycle there. The orbits of $\langle\sigma\rangle$
_are_ the cycles of $\sigma$. This proves that every permutation has a cycle
decomposition, unique up to reordering the cycles and rotating within each. For
instance, $\langle (1\,2)(3\,4\,5) \rangle$ acting on $\{1,\dots,5\}$ has orbits
$\{1,2\}$ and $\{3,4,5\}$.

$$
% caption: The cyclic group generated by $(1\,2)(3\,4\,5)$ splits $\{1,\dots,5\}$
% into two orbits; the generator runs each orbit as one cycle.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[draw=black, dashed] (-0.8,-1.2) rectangle (7.6,1.4);
\node[anchor=north west, font=\footnotesize, text=black] at (-0.75,1.35) {set $\{1,2,3,4,5\}$};
% orbit 1: {1,2}
\node[draw=acc, circle, fill=acc!10, minimum size=7mm] (n1) at (0,0) {$1$};
\node[draw=acc, circle, fill=acc!10, minimum size=7mm] (n2) at (1.8,0) {$2$};
\draw[->, acc, thick, bend left=35] (n1) to (n2);
\draw[->, acc, thick, bend left=35] (n2) to (n1);
\node[font=\footnotesize, text=acc] at (0.9,-0.95) {orbit A};
% orbit 2: {3,4,5}
\node[draw=acc, circle, fill=acc!10, minimum size=7mm] (n3) at (4.4,0.5) {$3$};
\node[draw=acc, circle, fill=acc!10, minimum size=7mm] (n4) at (6.2,0.9) {$4$};
\node[draw=acc, circle, fill=acc!10, minimum size=7mm] (n5) at (6.6,-0.5) {$5$};
\draw[->, acc, thick, bend left=20] (n3) to (n4);
\draw[->, acc, thick, bend left=20] (n4) to (n5);
\draw[->, acc, thick, bend left=20] (n5) to (n3);
\node[font=\footnotesize, text=acc] at (5.6,-1.0) {orbit B};
\end{tikzpicture}
$$

### 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
caption: $\textsc{Orbit}(a, S)$ — the orbit of $a$ under the group generated by $S$
initialize $O \gets \{a\}$ and a queue $Q \gets [a]$
while $Q$ is nonempty do
  dequeue $x$ from $Q$
  for each generator $s \in S$ do
    $y \gets s \cdot x$
    if $y \notin O$ then
      add $y$ to $O$ and enqueue $y$
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.

$$
% caption: $D_8$ acts transitively on the square's four vertices (one orbit of
% size $4$); the stabilizer of a vertex is the reflection through it, of order $2$.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\coordinate (v1) at (0,2);
\coordinate (v2) at (2,2);
\coordinate (v3) at (2,0);
\coordinate (v4) at (0,0);
\draw[draw=black, thick] (v1) -- (v2) -- (v3) -- (v4) -- cycle;
\foreach \p/\n in {v1/1, v2/2, v3/3, v4/4}
  \node[draw=acc, circle, fill=acc!10, minimum size=6mm] at (\p) {$\n$};
% rotation arrow
\draw[->, acc, thick] (2.7,1) arc (-50:50:0.7);
\node[font=\footnotesize, text=acc, anchor=west] at (3.15,1) {$r$};
% reflection axis through vertices 1 and 3 (interior segment only)
\draw[black, dashed] (0.5,1.5) -- (1.5,0.5);
\node[font=\footnotesize, text=black, anchor=north] at (1,-0.6) {dashed diagonal is the axis of $s$};
\node[font=\footnotesize, anchor=north] at (1,-1.1) {stabilizer of vertex $1$ has order $2$};
\end{tikzpicture}
$$

Transitivity is not inherited by subgroups: the subgroup $\langle (1\,2),
(3\,4)\rangle \le S_4$ 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**, $g \cdot a = ga$. 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 $ba^{-1}$ sends $a$ to $b$) and faithful (if $ga = a$ 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.[^df-cayley]

> **Theorem (Cayley).** Every group $G$ is isomorphic to a subgroup of a symmetric
> group. If $|G| = n$, then $G$ embeds in $S_n$.

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

> **Worked example.** Label the Klein four-group $V = \{1, a, b, c\}$ by
> $1,2,3,4$. Left multiplication by $a$ sends $1 \mapsto a$, $a \mapsto 1$,
> $b \mapsto c$, $c \mapsto b$, so $a$ acts as $(1\,2)(3\,4)$; likewise $b$ acts as
> $(1\,3)(2\,4)$ and $c$ as $(1\,4)(2\,3)$. Each nonidentity element becomes a
> fixed-point-free involution, and $V$ embeds in $S_4$ as the Klein four-group of
> double transpositions.

$$
% caption: The left regular representation of the Klein four-group: each element,
% acting by left multiplication, becomes a fixed-point-free permutation in $S_4$.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\node[font=\footnotesize, text=black] at (0,1.2) {element};
\node[font=\footnotesize, text=black] at (4.4,1.2) {its permutation in $S_4$};
\foreach \i/\g/\perm in {0/{1}/{\text{identity}}, 1/{a}/{(1\,2)(3\,4)}, 2/{b}/{(1\,3)(2\,4)}, 3/{c}/{(1\,4)(2\,3)}} {
  \node[draw=acc, fill=acc!10, minimum width=8mm, minimum height=6mm] at (0,-\i*0.85) {$\g$};
  \draw[->, acc] (0.6,-\i*0.85) -- (2.9,-\i*0.85);
  \node[anchor=west] at (3.0,-\i*0.85) {$\perm$};
}
\end{tikzpicture}
$$

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 $H \le G$ and let $G$ act on the set $A$ of left cosets of $H$ by
$g \cdot aH = (ga)H$.[^df-thm3]

> **Theorem (Action on cosets).** Let $H \le G$, let $G$ act by left
> multiplication on the left cosets of $H$, and let $\pi_H : G \to S_A$ be the
> associated representation. Then:
>
> - $G$ acts transitively on $A$;
> - the stabilizer of the coset $1H$ is $H$;
> - $\ker\pi_H = \bigcap_{x \in G} xHx^{-1}$, the **largest normal subgroup of $G$
>   contained in $H$**.

Transitivity holds because $ba^{-1}$ carries $aH$ to $bH$. The stabilizer of $1H$
is $\{g : gH = H\} = H$. For the kernel, $g$ fixes every coset $xH$ exactly when
$x^{-1}gx \in H$ for all $x$, i.e. $g \in \bigcap_x xHx^{-1}$; 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.

> **Corollary (Smallest-prime index is normal).** If $G$ is finite and $p$ is the
> smallest prime dividing $|G|$, then any subgroup $H$ of index $p$ is normal.

Let $K = \ker\pi_H$. Then $G/K$ embeds in $S_p$, so $[G:K]$ divides $p!$. Writing
$[H:K] = k$ gives $[G:K] = pk$, and $pk \mid p!$ forces $k \mid (p-1)!$. 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 $H = K \trianglelefteq
G$.[^df-cor5] 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 | $H$ | $\bigcap_x xHx^{-1}$ |
| 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](/abstract-algebra/group-actions-and-sylow/conjugation-and-the-class-equation):
$g \cdot a = gag^{-1}$ replaces multiplication, the orbits become conjugacy
classes, and orbit-stabilizer applied to it produces the class equation.

[^df-intro]: **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.
[^df-prop1]: **Dummit & Foote**, §4.1, Proposition 1: the bijection between actions of $G$ on $A$ and homomorphisms $G \to S_A$.
[^df-prop2]: **Dummit & Foote**, §4.1, Proposition 2: the orbit relation is an equivalence relation and $|\mathcal{O}_a| = [G:G_a]$, together with the cycle-decomposition application.
[^df-cayley]: **Dummit & Foote**, §4.2, Corollary 4: Cayley's theorem via the left regular representation.
[^df-thm3]: **Dummit & Foote**, §4.2, Theorem 3: the action on left cosets of $H$, its transitivity, stabilizer, and kernel as the largest normal subgroup inside $H$.
[^df-cor5]: **Dummit & Foote**, §4.2, Corollary 5: a subgroup of index equal to the smallest prime divisor of $|G|$ is normal.