Homomorphisms, Isomorphisms, and Actions

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 and matrix 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

With the operations suppressed, the condition reads $φ (x y) = φ (x) φ (y)$ , 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, $φ (1_{G}) = 1_{H}$ and $φ (x^{- 1}) = φ (x)^{- 1}$ , and by induction $φ (x^{n}) = φ (x)^{n}$ for every integer $n$ .¹

Isomorphisms

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 $≅$ is an equivalence relation whose classes are the true objects of study.¹ 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. $∣ x ∣ = ∣ φ (x) ∣$ for every $x$ , so $G$ and $H$ have the same number of elements of each order.

For example, $(R ∖ {0}, \times)$ and $(R, +)$ are not isomorphic: the first has an element of order $2$ , namely $- 1$ , and the second has none. The exponential map $exp : (R, +) \to (R^{> 0}, \times)$ , 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.

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.

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 $D_{6} ≅ S_{3}$ and $GL_{2} (F_{2}) ≅ S_{3}$ without constructing explicit maps, and up to isomorphism there are exactly two groups of order $6$ , namely $S_{3}$ and $Z /6 Z$ .¹

Kernel and image

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

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.

When $φ$ is injective, $G ≅ φ (G)$ , 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. The bijections $G \to G$ that are isomorphisms form a group under composition, the automorphism group $Aut (G)$ , studied in automorphisms and simplicity.

Group actions

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

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 $σ_{g} : A \to A$ by $σ_{g} (a) = g \cdot a$ , and this map is a permutation of $A$ : its two-sided inverse is $σ_{g^{- 1}}$ , since $σ_{g^{- 1}} (σ_{g} (a)) = g^{- 1} \cdot (g \cdot a) = (g^{- 1} g) \cdot a = a$ .²

Actions are homomorphisms into $S_{A}$

Collecting the permutations $σ_{g}$ gives a map $G \to S_{A}$ , and the first action axiom makes it a homomorphism.

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.

The standard actions

Four actions recur throughout the theory.²

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 $g \cdot a = g a$ . By cancellation this is faithful, and it underlies Cayley's theorem.
Symmetry action. $D_{2 n}$ 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 $D_{6} ≅ S_{3}$ .
Conjugation. $G$ acts on itself by $g \cdot a = g a g^{- 1}$ . Each map $a \mapsto g a g^{- 1}$ is an automorphism, the source of the class equation.

Orbits and stabilizers

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

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 $a$ .² 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; the geometric intuition is already visible in a rotation acting on a square.

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.

The two ideas together

	Homomorphism $φ : G \to H$	Action of $G$ on $A$
Data	respects the operation	$φ : G \to S_{A}$ (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	$φ$ injective	$ker = {1}$ , so $G ↪ S_{A}$
Key subgroups	kernel, image	kernel, stabilizers
Partition induced	fibers of $φ$	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.

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. ↩ ↩² ↩³ ↩⁴
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. ↩ ↩² ↩³ ↩⁴

Homomorphisms

Isomorphisms

Kernel and image

Group actions

Actions are homomorphisms into SA​

The standard actions

Orbits and stabilizers

The two ideas together

Footnotes

Actions are homomorphisms into $S_{A}$