group actions群作用ぐんさよう and symmetry対称性たいしょうせい

date2026-07-14document_iddoc_df5fca749526db988e1d75f3584d1a34description群作用を、群を集合上の変換として見る方法として導入し、軌道・固定部分群・対称性を説明する。prerequisites[群/ぐん]の[基本/きほん] / [群準同型/ぐんじゅんどうけい]と[同型/どうけい] / [剰余類/じょうよるい]とラグランジュの[定理/ていり] / [写像/しゃぞう]の[基本/きほん]type講義content_typelecturestatusactive

mathabstract-algebragroup-theorysymmetrylecture

A group群ぐん can be viewed not only as an operation table演算表えんざんひょう, but also as a collection of transformations変換へんかん that move a set集合しゅうごう. This viewpoint is a group action群作用ぐんさよう.

Using group actions, we can directly describe how elements of a group move objects. Basic examples are the action of a symmetry group on a figure, a permutation group on its underlying set, and a group of invertible matrices through linear transformations.

data/lecture/math/linear-algebra/linear-maps-and-matrices.lecture.n.md

Linear transformations are included only as an application preview. The definitions and proofs on this page use only transformations that move elements of a set to other elements.

1Definition of a group action

A group $G$ acts on a set $X$ if a map $G \times X \to X$ , $(g, x) \mapsto g \cdot x$ , is given and the following conditions hold for every $g, h \in G$ and $x \in X$ .

e \cdot x = x

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

The first equation says that the identity element単位元たんいげん does nothing. The second says that multiplication乗法じょうほう in the group corresponds to composition合成ごうせい of transformations.

Each $g \in G$ defines a bijection $x \mapsto g \cdot x$ of $X$ , whose inverse is the action of $g^{- 1}$ . If $Sym (X)$ denotes the group of all bijections from $X$ to itself, a group action is equivalently a group homomorphism

G \to Sym (X), g \mapsto (x \mapsto g \cdot x) .

Indeed, if $ρ (g) (x) = g \cdot x$ , then

ρ (g h) (x) = (g h) \cdot x = g \cdot (h \cdot x) = (ρ (g) \circ ρ (h)) (x),

so $ρ$ is a homomorphism. Conversely, given a homomorphism $ρ : G \to Sym (X)$ , define $g \cdot x = ρ (g) (x)$ . The identities $ρ (e) = {id}_{X}$ and $ρ (g h) = ρ (g) \circ ρ (h)$ give the two action axioms.

2Orbits

The orbit軌道きどう of an element $x \in X$ is defined by

G x = {g \cdot x ∣ g \in G}

The orbit is the set of all elements reachable from $x$ by the action作用さよう of the group.

Define $y \sim x$ to mean that $y = g \cdot x$ for some $g \in G$ . This is an equivalence relation. Reflexivity follows from $x = e \cdot x$ ; symmetry follows because $y = g \cdot x$ implies $x = g^{- 1} \cdot y$ ; and transitivity follows because $y = g \cdot x$ and $z = h \cdot y$ imply $z = (h g) \cdot x$ . Thus the orbits partition $X$ into disjoint equivalence classes.

3Stabilizers

The set of group elements that do not move $x \in X$ ,

G_{x} = {g \in G ∣ g \cdot x = x}

is called the stabilizer. It is a subgroup of $G$ . Indeed, it is nonempty because $e \cdot x = x$ . If $g, h \in G_{x}$ , then

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

Here applying $h^{- 1}$ to $h \cdot x = x$ gives $h^{- 1} \cdot x = x$ . The subgroup criterion therefore gives $G_{x} [PARSE ERROR: Undefined("Command(\"le\")")] G$ .

The orbit describes where $x$ can move, and the stabilizer describes which group elements fix $x$ .

4Kernel, image, and faithfulness of an action

Let $ρ : G \to Sym (X)$ be the homomorphism corresponding to an action. Then

\ker ρ = {g \in G ∣ g \cdot x = x forevery x \in X} = ⋂_{x \in X} G_{x} .

The first equality restates when $ρ (g)$ is the identity map, and the second restates when $g$ belongs to every stabilizer. Thus $\ker ρ$ is normal, while $Im ρ [PARSE ERROR: Undefined("Command(\"le\")")] Sym (X)$ is the subgroup of transformations of $X$ that actually occur.

An action is faithful when $\ker ρ = {e}$ , equivalently when $ρ$ is injective. Faithfulness does not require every stabilizer $G_{x}$ to be trivial; the action on the vertices of an equilateral triangle is an example.

5Concrete example: symmetries of an equilateral triangle

Let $X = {1, 2, 3}$ be the vertex set of an equilateral triangle. The rotational and reflection symmetries of the triangle permute the vertices. Therefore the symmetry group acts on $X$ .

The orbit of vertex 1 is ${1, 2, 3}$ , because a symmetry can move vertex 1 to any vertex. On the other hand, the symmetries that fix vertex 1 are the identity transformation and the reflection across the axis passing through vertex 1.

This action is faithful: the identity is the only symmetry that fixes all three vertices, so the kernel of the action is ${e}$ . Nevertheless, the stabilizer of vertex 1 also contains a reflection. This example distinguishes fixing every vertex simultaneously from fixing one vertex.

6What changes and what is preserved

In a group action群作用ぐんさよう, elements of the group are viewed as transformations. Under the action, an element of the set may move to another element of the same set. What is preserved is the structure that multiplication in the group corresponds to composition of transformations.

The idea of group actions appears widely in geometry幾何きか, linear algebra線型代数せんけいだいすう, physics物理ぶつり, and combinatorics組合せ論.

7Theorem: orbit-stabilizer theorem

Let a group $G$ act on a set $X$ , and let $x \in X$ . There is a bijection between the set of left cosets of $G_{x}$ and the orbit $G x$ , so

| G x | = [G : G_{x}]

The stabilizer $G_{x}$ need not be normal, so $G / G_{x}$ here denotes a set of left cosets, not a quotient group. In particular, when $G$ is finite,

| G | = | G x | | G_{x} |

This is the orbit-stabilizer theorem軌道固定部分群定理きどうこていぶぶんぐんていり.

For the symmetry group of an equilateral triangle, the orbit of vertex 1 has 3 elements, and the stabilizer has 2 elements. Hence the order位数いすう of the group is $3 \cdot 2 = 6$ .

8Proof supplement: why the orbit-stabilizer theorem holds

Suppose a group群ぐん $G$ acts on a set $X$ , and fix $x \in X$ . Write the stabilizer固定部分群こていぶぶんぐん of $x$ as $G_{x} = {g \in G ∣ g \cdot x = x}$ .

Here $G / G_{x}$ denotes the set of left cosets. Consider the map

G / G_{x} \to G \cdot x, g G_{x} \mapsto g \cdot x .

We check that this map is well-defined. If $g G_{x} = h G_{x}$ , then $h^{- 1} g \in G_{x}$ . Therefore

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

Acting by $h$ on the left gives $g \cdot x = h \cdot x$ . Hence changing the representative does not change the output.

Conversely, if $g \cdot x = h \cdot x$ , then $(h^{- 1} g) \cdot x = x$ , so $h^{- 1} g \in G_{x}$ and therefore $g G_{x} = h G_{x}$ . Thus the map is injective. Every element of $G \cdot x$ has the form $g \cdot x$ by definition, so the map is also surjective. For a finite group,

| G \cdot x | = [G : G_{x}] = \frac{| G |}{| G_{x} |}

follows.

9Exercise link

data/exercise/math/abstract-algebra/group-actions-and-orbit-stabilizer.exercise.n.md

10Summary

A group action views a group as transformations of a set through a homomorphism $G \to Sym (X)$ . Orbits partition $X$ by reachability, while the stabilizer $G_{x}$ collects the group elements that fix $x$ . The orbit-stabilizer theorem connects these notions through left cosets.

This completes our study of groups, which have one operation, and of how groups act on sets. We next turn to rings, which carry two operations: addition and multiplication.