Group action

Intuitive definition

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.

Formal definition

Let $G$ be a group with identity $e$ and $X$ a non-empty set. A (left) action of $G$ on $X$ is a map
$$G \times X \to X, \qquad (g, x) \mapsto g \cdot x$$satisfying the two axioms:

1. $e \cdot x = x$ for all $x \in X$;
2. $g \cdot (h \cdot x) = (gh) \cdot x$ for all $g, h \in G$ and $x \in X$.

We say that $G$ acts (or operates) on $X$, that $X$ is a $G$-set, and we write $G \curvearrowright X$.

Equivalent point of view. Giving an action of $G$ on $X$ amounts to giving a group homomorphism
$$\tau : G \to \mathfrak{S}(X), \qquad \tau(g)(x) = g \cdot x,$$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.

Examples

1. Natural action of the symmetric group. $\mathfrak{S}_n$ acts on $\{1, \ldots, n\}$ by $\sigma \cdot i = \sigma(i)$.
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 $g \cdot (aH) = (ga)H$.
3. Conjugation. $G$ acts on itself by $g \cdot x = g x g^{-1}$, and on the set of its subgroups by $g \cdot H = g H g^{-1}$.
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 $P \cdot S = P S P^{\top}$.
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.
6. Trivial action. $g \cdot x = x$ for all $g$ and $x$.

Remarks

• 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 $\sigma \cdot (c_1, \ldots, c_n) = (c_{\sigma^{-1}(1)}, \ldots, c_{\sigma^{-1}(n)})$.
• 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 $g \cdot x = x \cdot g^{-1}$.
• 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.