Group action
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9/27/2026, 12:26:19 PM · visitor
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An action of a [[Group|group]] $G$ on a set $X$ is a map1
## Intuitive definition2
$$\begin{aligned}2
A group $G$ acts on a set $X$ when every element of $G$ "moves" the points of $X$ in a way compatible with the group law. The action creates a two-way dialogue: from $G$ to $X$, it cuts $X$ into orbits (classification, counting); from $X$ to $G$, it produces natural subgroups (stabilizers, kernel) and embeddings of $G$ into permutation groups.3
G \times X &\longrightarrow X \\3
4
(g, x) &\longmapsto g \cdot x4
## Formal definition5
\end{aligned}$$5
Let $G$ be a group with identity $e$ and $X$ a non-empty set. A (left) **action** of $G$ on $X$ is a map6
verifying the following properties for all $x\in X$: 6
$$G \times X \to X, \qquad (g, x) \mapsto g \cdot x$$7
1) Action of the identity: $e\cdot x=x$7
satisfying the two axioms:8
2) Successive actions : $(gh)\cdot x= g\cdot (h\cdot x)$ for all $g,h\in G$.8
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1. $e \cdot x = x$ for all $x \in X$;10
2. $g \cdot (h \cdot x) = (gh) \cdot x$ for all $g, h \in G$ and $x \in X$.11
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We say that $G$ **acts** (or **operates**) on $X$, that $X$ is a **$G$-set**, and we write $G \curvearrowright X$.13
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**Equivalent point of view.** Giving an action of $G$ on $X$ amounts to giving a group homomorphism15
$$\tau : G \to \mathfrak{S}(X), \qquad \tau(g)(x) = g \cdot x,$$16
where $\mathfrak{S}(X)$ is the group of bijections of $X$. Indeed, axiom 2 says $\tau(gh) = \tau(g) \circ \tau(h)$ and axiom 1 says $\tau(e) = \mathrm{id}_X$; hence each $\tau(g)$ is a bijection with inverse $\tau(g^{-1})$. The first point of view is the more intuitive one; the second brings out the kernel $\ker \tau$ and connects actions with permutation groups.17
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## Examples19
1. **Natural action of the symmetric group.** $\mathfrak{S}_n$ acts on $\{1, \ldots, n\}$ by $\sigma \cdot i = \sigma(i)$.20
2. **Left translation.** $G$ acts on itself by $g \cdot x = gx$. More generally, if $H$ is a subgroup of $G$, then $G$ acts on the set $G/H$ of left cosets by $g \cdot (aH) = (ga)H$.21
3. **Conjugation.** $G$ acts on itself by $g \cdot x = g x g^{-1}$, and on the set of its subgroups by $g \cdot H = g H g^{-1}$.22
4. **Linear actions.** $\mathrm{GL}_n(K)$ acts on $K^n$ by $A \cdot v = Av$; on $M_n(K)$ by conjugation $P \cdot A = PAP^{-1}$; on symmetric matrices by congruence $P \cdot S = P S P^{\top}$.23
5. **Geometry.** The dihedral group $D_4$ of isometries of the square acts on its $4$ vertices, on its $4$ edges, on its $2$ diagonals: one group, very different actions.24
6. **Trivial action.** $g \cdot x = x$ for all $g$ and $x$.25
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## Remarks27
- **Classical pitfall.** One would like $\mathfrak{S}_n$ to act on $K^n$ by permuting coordinates, $\sigma \cdot (c_1, \ldots, c_n) = (c_{\sigma(1)}, \ldots, c_{\sigma(n)})$. This is not a left action: one checks that $\sigma \cdot (\tau \cdot c) = (\tau\sigma) \cdot c$, the order is reversed. The correct formula is $\sigma \cdot (c_1, \ldots, c_n) = (c_{\sigma^{-1}(1)}, \ldots, c_{\sigma^{-1}(n)})$.28
- A **right action** is a map $X \times G \to X$ with $(x \cdot g) \cdot h = x \cdot (gh)$; it becomes a left action by setting $g \cdot x = x \cdot g^{-1}$.29
- Before speaking of an action, check that $g \cdot x$ really lies in $X$: for instance $\mathfrak{S}_n$ acts on the $m$-element subsets of $\{1, \ldots, n\}$ because a bijection preserves cardinality.Revision 5981
9/27/2026, 12:25:16 PM · visitor
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bibliographyNonePerrin, Daniel — Cours d’algèbre (Ellipses) — Chap. I
Revision 532
7/20/2026, 8:43:19 PM · Ancient Tree
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An action of a group $G$ on a set $X$ is a map1
An action of a [[Group|group]] $G$ on a set $X$ is a map2
$$\begin{aligned}3
G \times X &\longrightarrow X \\4
(g, x) &\longmapsto g \cdot x5
\end{aligned}$$6
verifying the following properties for all $x\in X$: 7
1) Action of the identity: $e\cdot x=x$8
2) Successive actions : $(gh)\cdot x= g\cdot (h\cdot x)$ for all $g,h\in G$.Revision 531
7/20/2026, 8:43:05 PM · Ancient Tree
Concept edited
Compare with revision 5239 changed lines
1
A group action is...1
An action of a group $G$ on a set $X$ is a map2
$$\begin{aligned}3
G \times X &\longrightarrow X \\4
(g, x) &\longmapsto g \cdot x5
\end{aligned}$$6
verifying the following properties for all $x\in X$: 7
1) Action of the identity: $e\cdot x=x$8
2) Successive actions : $(gh)\cdot x= g\cdot (h\cdot x)$ for all $g,h\in G$.Revision 523
7/20/2026, 8:19:25 PM · Ancient Tree
Concept created
A group action is...