---
title: Homomorphisms, Isomorphisms, and Actions
module: Groups and Symmetry
moduleNumber: 2
lessonNumber: 4
order: 204
summary: >
  A homomorphism is a map between groups that respects the operation; an
  isomorphism is a bijective one, making two groups the same up to relabeling.
  The kernel and image measure how far a homomorphism is from injective and
  surjective. A group action realizes a group as permutations of a set, and
  actions correspond exactly to homomorphisms into a symmetric group, with orbits
  and stabilizers as the first tools for counting.
topics: [Groups and Symmetry]
sources:
  - book: Dummit & Foote
    ref: "Ch. 1 — Introduction to Groups; §1.6 Homomorphisms and Isomorphisms, §1.7 Group Actions"
draft: false
---

Two questions the group axioms alone cannot settle: when are two groups the same,
and what does it mean for a group to act on a set? The
[dihedral](/abstract-algebra/groups-and-symmetry/dihedral-and-symmetric-groups)
and [matrix](/abstract-algebra/groups-and-symmetry/matrix-and-quaternion-groups)
groups already appeared under several guises, so a precise notion of sameness is
overdue. A homomorphism compares two groups through a structure-preserving map; a
group action realizes a group's elements as concrete symmetries. The two ideas
coincide: an action is a homomorphism into a symmetric group.

## Homomorphisms

> **Definition (Homomorphism).** A **homomorphism** from a group $(G, \ast)$ to a
> group $(H, \diamond)$ is a map $\varphi : G \to H$ satisfying
> $\varphi(x \ast y) = \varphi(x) \diamond \varphi(y)$ for all $x, y \in G$.

With the operations suppressed, the condition reads $\varphi(xy) = \varphi(x)\varphi(y)$,
where the product on the left is taken in $G$ and the product on the right in $H$.
A homomorphism carries the group structure across: from the defining equation,
$\varphi(1_G) = 1_H$ and $\varphi(x^{-1}) = \varphi(x)^{-1}$, and by induction
$\varphi(x^n) = \varphi(x)^n$ for every integer $n$.[^df-hom]

### Isomorphisms

> **Definition (Isomorphism).** An **isomorphism** is a bijective homomorphism.
> Groups $G$ and $H$ are **isomorphic**, written $G \cong H$, if there is an
> isomorphism between them.

Isomorphic groups are the same group with the elements and operation possibly
written differently. Any property expressible from the axioms alone transfers
across an isomorphism, so $\cong$ is an equivalence relation whose classes are
the true objects of study.[^df-hom] Two consequences give quick tests that two
groups are _not_ isomorphic.

- **Cardinality.** $|G| = |H|$, since an isomorphism is a bijection.
- **Commutativity.** $G$ is abelian if and only if $H$ is.
- **Order spectrum.** $|x| = |\varphi(x)|$ for every $x$, so $G$ and $H$ have the
  same number of elements of each order.

For example, $(\mathbb{R} \setminus \{0\}, \times)$ and $(\mathbb{R}, +)$ are not
isomorphic: the first has an element of order $2$, namely $-1$, and the second has
none. The exponential map $\exp : (\mathbb{R}, +) \to (\mathbb{R}^{>0}, \times)$,
by contrast, _is_ an isomorphism, since $e^{x + y} = e^x e^y$ and $\exp$ is a
bijection with inverse $\log$. The elements and operations look different; the
groups are identical.

$$
% caption: An isomorphism is a relabeling that carries one Cayley table onto
% another; sending 0,1,2,3 to the powers of a rotation matches every entry, so
% the additive group of residues mod four and the rotation group of the square
% are the same group.
\begin{tikzpicture}[font=\small, x=10mm, y=10mm]
  \definecolor{acc}{HTML}{2A6F97}
  % left table Z/4
  \begin{scope}[shift={(-7,2.5)}]
    \node at (0.5,-0.5) {$+$};
    \foreach \v [count=\c from 1] in {0,1,2,3} {
      \node[acc] at (\c+0.5,-0.5) {$\v$};
      \node[acc] at (0.5,-\c-0.5) {$\v$};
    }
    \foreach \i in {0,1,2,3}
      \foreach \j in {0,1,2,3} {
        \pgfmathtruncatemacro{\s}{mod(\i+\j,4)}
        \node at (\j+1.5,-\i-1.5) {$\s$};
      }
    \draw (0,0) grid (5,-5);
    \node[font=\footnotesize, anchor=north] at (2.5,-5.2) {residues mod 4};
  \end{scope}
  % arrow
  \draw[->, acc, very thick] (-1.3,0) -- (0.2,0) node[midway, above, font=\footnotesize] {relabel};
  % right table powers of r
  \begin{scope}[shift={(1,2.5)}]
    \foreach \v [count=\c from 1] in {{r^0},{r^1},{r^2},{r^3}} {
      \node[acc] at (\c+0.5,-0.5) {$\v$};
      \node[acc] at (0.5,-\c-0.5) {$\v$};
    }
    \foreach \i in {0,1,2,3}
      \foreach \j in {0,1,2,3} {
        \pgfmathtruncatemacro{\s}{mod(\i+\j,4)}
        \node at (\j+1.5,-\i-1.5) {$r^{\s}$};
      }
    \draw (0,0) grid (5,-5);
    \node[font=\footnotesize, anchor=north] at (2.5,-5.2) {rotations of the square};
  \end{scope}
\end{tikzpicture}
$$

Deciding whether an isomorphism exists is generally hard; the work is to exhibit
one or to prove none can exist. Classification theorems do this in bulk. One
early result:
every nonabelian group of order $6$ is isomorphic to $S_3$, so $D_6 \cong S_3$ and
$\mathrm{GL}_2(\mathbb{F}_2) \cong S_3$ without constructing explicit maps, and up
to isomorphism there are exactly two groups of order $6$, namely $S_3$ and
$\mathbb{Z}/6\mathbb{Z}$.[^df-hom]

### Kernel and image

Two subgroups measure how a general homomorphism departs from being an
isomorphism.

> **Definition (Kernel and image).** For a homomorphism $\varphi : G \to H$, the
> **kernel** is $\ker\varphi = \{g \in G : \varphi(g) = 1_H\}$, the fiber over the
> identity, and the **image** is $\varphi(G) = \{\varphi(g) : g \in G\}$.

> **Proposition.** $\ker\varphi$ is a subgroup of $G$, $\varphi(G)$ is a subgroup
> of $H$, and $\varphi$ is injective if and only if $\ker\varphi = \{1_G\}$.

> **Proof of the injectivity criterion.** If $\varphi$ is injective, only $1_G$ can
> map to $1_H$, so the kernel is trivial. Conversely, if $\ker\varphi = \{1_G\}$ and
> $\varphi(x) = \varphi(y)$, then $\varphi(xy^{-1}) = \varphi(x)\varphi(y)^{-1} = 1_H$,
> so $xy^{-1} \in \ker\varphi = \{1_G\}$, giving $x = y$.[^df-hom] The kernel is the
> exact obstruction to injectivity: elements sharing a value differ by a kernel
> element, so the fibers of $\varphi$ are all translates of $\ker\varphi$ and all
> have the same size.

$$
% caption: A homomorphism folds its domain onto the image; the fiber over the
% identity is the kernel, and every other fiber is a translate of it, so all
% fibers have the same size and the map is injective exactly when the kernel is
% trivial.
\begin{tikzpicture}[font=\small, >=stealth]
  \definecolor{acc}{HTML}{2A6F97}
  % domain oval with three fibers
  \draw[thick] (-4.3,-1.7) ellipse (1.5 and 2.2);
  \node[font=\footnotesize] at (-4.3,0.9) {$G$};
  \foreach \y/\lab in {-0.6/{}, -1.7/{}, -2.8/{}} {
    \draw[black] (-5.2,\y+0.35) rectangle (-3.4,\y-0.35);
  }
  \draw[acc, thick] (-5.2,-1.35) rectangle (-3.4,-2.05);
  \node[acc, font=\footnotesize, anchor=east] at (-5.35,-1.7) {kernel};
  % codomain oval with three points
  \draw[thick] (1.5,-1.7) ellipse (1.2 and 2.2);
  \node[font=\footnotesize] at (1.5,0.9) {$H$};
  \node[circle, fill=black, inner sep=1.4pt] (t1) at (1.5,-0.6) {};
  \node[circle, fill=acc, inner sep=1.6pt] (t2) at (1.5,-1.7) {};
  \node[circle, fill=black, inner sep=1.4pt] (t3) at (1.5,-2.8) {};
  \node[acc, font=\footnotesize, anchor=west] at (2.0,-1.7) {$1_H$};
  % arrows
  \draw[->, black] (-3.3,-0.6) -- (t1);
  \draw[->, acc] (-3.3,-1.7) -- (t2);
  \draw[->, black] (-3.3,-2.8) -- (t3);
  \node[acc, font=\footnotesize] at (-0.9,0.05) {homomorphism};
\end{tikzpicture}
$$

When $\varphi$ is injective, $G \cong \varphi(G)$, so $G$ is realized as a
subgroup of $H$. This is how abstract groups get concrete descriptions, and the
extreme case — every group embeds in a symmetric group — is
[Cayley's theorem](/abstract-algebra/group-actions-and-sylow/actions-and-cayleys-theorem).
The bijections $G \to G$ that are isomorphisms form a group under composition, the
**automorphism group** $\mathrm{Aut}(G)$, studied in
[automorphisms and simplicity](/abstract-algebra/group-actions-and-sylow/automorphisms-and-simple-groups).

> **Worked example.** The sign map $\varepsilon : S_3 \to \{\pm 1\}$, sending even
> permutations to $+1$ and odd ones to $-1$, is a homomorphism, since the parity of
> a product is the sum of the parities. The three transpositions
> $(1\ 2), (1\ 3), (2\ 3)$ map to $-1$; the identity and the two $3$-cycles
> $(1\ 2\ 3), (1\ 3\ 2)$ map to $+1$. So
>
> $$
> \ker\varepsilon = \{e,\ (1\ 2\ 3),\ (1\ 3\ 2)\} = A_3, \qquad \varepsilon(S_3) = \{\pm 1\}.
> $$
>
> The kernel has order $3$ and the image order $2$, with $3 \cdot 2 = 6 = |S_3|$:
> the kernel's index equals the size of the image, the pattern the
> [isomorphism theorems](/abstract-algebra/subgroups-and-quotients/isomorphism-theorems)
> make general.

## Group actions

An action lets a group operate on a set, permuting its elements in a way
compatible with the group operation.

> **Definition (Group action).** A **(left) action** of a group $G$ on a set $A$
> is a map $G \times A \to A$, written $(g, a) \mapsto g \cdot a$, such that
>
> - $g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a$ for all $g_1, g_2 \in G$ and $a \in A$, and
> - $1 \cdot a = a$ for all $a \in A$.

The two axioms say that acting by $g_2$ then $g_1$ equals acting by the product
$g_1 g_2$, and that the identity does nothing. From them, each fixed $g$ defines a
map $\sigma_g : A \to A$ by $\sigma_g(a) = g \cdot a$, and this map is a
**permutation** of $A$: its two-sided inverse is $\sigma_{g^{-1}}$, since
$\sigma_{g^{-1}}(\sigma_g(a)) = g^{-1} \cdot (g \cdot a) = (g^{-1}g)\cdot a = a$.[^df-act]

### Actions are homomorphisms into $S_A$

Collecting the permutations $\sigma_g$ gives a map $G \to S_A$, and the first
action axiom makes it a homomorphism.

> **Theorem (Permutation representation).** For an action of $G$ on $A$, the map
> $\varphi : G \to S_A$ defined by $\varphi(g) = \sigma_g$ is a homomorphism,
> called the **permutation representation** of the action. Conversely, any
> homomorphism $\varphi : G \to S_A$ defines an action by $g \cdot a = \varphi(g)(a)$.
> Actions of $G$ on $A$ and homomorphisms $G \to S_A$ are the same data.

> **Proof that $\varphi$ is a homomorphism.** For all $a \in A$,
>
> $$
> \sigma_{g_1 g_2}(a) = (g_1 g_2)\cdot a = g_1 \cdot (g_2 \cdot a)
>   = \sigma_{g_1}(\sigma_{g_2}(a)) = (\sigma_{g_1} \circ \sigma_{g_2})(a),
> $$
>
> so $\varphi(g_1 g_2) = \varphi(g_1)\varphi(g_2)$ as permutations.[^df-act] The
> correspondence is the reason group actions and homomorphisms into symmetric groups
> are treated interchangeably.

$$
% caption: An action names each group element as a permutation of the set; the
% permutation representation is the resulting homomorphism into the symmetric
% group, and its kernel is the set of elements that fix every point.
\begin{tikzpicture}[font=\small, >=stealth,
  b/.style={draw, minimum width=20mm, minimum height=12mm, align=center}]
  \definecolor{acc}{HTML}{2A6F97}
  \node[b, draw=acc] (g) at (0,0) {group $G$};
  \node[b] (s) at (6,0) {$S_A$};
  \draw[->, acc, thick] (g) -- (s)
    node[midway, above, font=\footnotesize] {each $g$ acts}
    node[midway, below, font=\footnotesize] {as a permutation};
  \node[font=\footnotesize, anchor=west] at (7,0) {kernel: moves no point of $A$};
\end{tikzpicture}
$$

> **Definition (Faithful action; kernel).** The **kernel** of an action is
> $\{g \in G : g \cdot a = a \text{ for all } a \in A\}$, equal to the kernel of
> the permutation representation. The action is **faithful** if distinct group
> elements induce distinct permutations, i.e. the kernel is $\{1\}$ and $\varphi$
> is injective.

### The standard actions

Four actions recur throughout the theory.[^df-act]

- **Trivial action.** $g \cdot a = a$ for all $g$: every element acts as the
  identity permutation. The kernel is all of $G$, so the action is unfaithful when
  $|G| > 1$.
- **Left regular action.** $G$ acts on itself by $g \cdot a = ga$. By
  cancellation this is faithful, and it underlies Cayley's theorem.
- **Symmetry action.** $D_{2n}$ acts on the $n$ vertices of the polygon; distinct
  symmetries permute the vertices differently, so the action is faithful. For
  $n = 3$ this gives an injective map $D_6 \to S_3$ between groups of equal order,
  hence $D_6 \cong S_3$.
- **Conjugation.** $G$ acts on itself by $g \cdot a = gag^{-1}$. Each map
  $a \mapsto gag^{-1}$ is an automorphism, the source of the
  [class equation](/abstract-algebra/group-actions-and-sylow/conjugation-and-the-class-equation).

### Orbits and stabilizers

An action carves the set into pieces and attaches a subgroup to each point.

> **Definition (Orbit and stabilizer).** For an action of $G$ on $A$ and a point
> $a \in A$:
>
> - the **orbit** of $a$ is $G \cdot a = \{g \cdot a : g \in G\}$, everything $a$
>   can be moved to;
> - the **stabilizer** of $a$ is $G_a = \{g \in G : g \cdot a = a\}$, the elements
>   that fix $a$.

The relation "$a \sim b$ if $b = g \cdot a$ for some $g$" is an equivalence
relation, so the orbits **partition** $A$, each an equivalence class. The
stabilizer $G_a$ is a subgroup of $G$ for every $a$.[^df-act] The interplay
between the size of an orbit and the size of its stabilizer is the
orbit–stabilizer theorem, developed in full in
[actions, orbits, and Cayley's theorem](/abstract-algebra/group-actions-and-sylow/actions-and-cayleys-theorem);
the geometric intuition is already visible in a rotation acting on a square.

$$
% caption: The four rotations of the square move every corner to every other, so
% the four vertices lie in a single orbit; the only rotation fixing a given
% corner is the identity, so each stabilizer is trivial.
\begin{tikzpicture}[font=\small, >=stealth]
  \definecolor{acc}{HTML}{2A6F97}
  \coordinate (a) at (-1.6,1.6);
  \coordinate (b) at (1.6,1.6);
  \coordinate (c) at (1.6,-1.6);
  \coordinate (d) at (-1.6,-1.6);
  \draw[thick] (a) -- (b) -- (c) -- (d) -- cycle;
  \foreach \p in {a,b,c,d}
    \node[circle, fill=acc, inner sep=2.2pt] at (\p) {};
  \node[above left] at (a) {$1$};
  \node[above right] at (b) {$2$};
  \node[below right] at (c) {$3$};
  \node[below left] at (d) {$4$};
  % rotation sends each corner to the next, tracing a four-cycle
  \draw[->, acc, thick] ($(a)+(0.4,-0.3)$)  to[bend right=12] ($(b)+(-0.4,-0.3)$);
  \draw[->, acc, thick] ($(b)+(-0.3,-0.4)$) to[bend right=12] ($(c)+(-0.3,0.4)$);
  \draw[->, acc, thick] ($(c)+(-0.4,0.3)$)  to[bend right=12] ($(d)+(0.4,0.3)$);
  \draw[->, acc, thick] ($(d)+(0.3,0.4)$)   to[bend right=12] ($(a)+(0.3,-0.4)$);
  \node[acc, font=\footnotesize] at (0,0) {one orbit: all four corners};
  \node[font=\footnotesize, anchor=north] at (0,-2.05) {stabilizer of each corner is trivial};
\end{tikzpicture}
$$

> **Worked example.** Let $D_8$, the symmetries of a square, act on its four
> vertices $\{1, 2, 3, 4\}$. The four rotations already send vertex $1$ to each
> position, so the orbit of $1$ is all of $\{1, 2, 3, 4\}$ and the action is
> transitive. The symmetries fixing vertex $1$ are the identity and the reflection
> across the diagonal through $1$ and $3$, so the stabilizer $G_1$ has order $2$.
> Then
>
> $$
> \lvert G_1\rvert \cdot \lvert G \cdot 1\rvert = 2 \cdot 4 = 8 = \lvert D_8\rvert,
> $$
>
> the orbit–stabilizer relation in a single case.

## The two ideas together

| | Homomorphism $\varphi : G \to H$ | Action of $G$ on $A$ |
| --- | --- | --- |
| Data | respects the operation | $\varphi : G \to S_A$ (a homomorphism) |
| Measures | how $G$ maps into $H$ | how $G$ permutes $A$ |
| "Trivial" case | image is $\{1_H\}$ | every $g$ fixes every point |
| "Faithful" case | $\varphi$ injective | $\ker = \{1\}$, so $G \hookrightarrow S_A$ |
| Key subgroups | kernel, image | kernel, stabilizers |
| Partition induced | fibers of $\varphi$ | orbits of the action |

An action is a homomorphism whose target is a symmetric group, so the two columns
are one theory read two ways.

[^df-hom]: **Dummit & Foote**, _Abstract Algebra_, §1.6 — homomorphisms and isomorphisms, the properties preserved by an isomorphism ($|G| = |H|$, abelian, order of elements), the classification of groups of order $6$, the kernel and image as subgroups, and the injectivity-by-trivial-kernel criterion.
[^df-act]: **Dummit & Foote**, _Abstract Algebra_, §1.7 — group actions, the permutation representation, the correspondence between actions and homomorphisms into $S_A$, faithful actions and the kernel of an action, the standard examples (trivial, left regular, dihedral, conjugation), and orbits and stabilizers.