A group action interprets group elements as invertible transformations of a set. It is the common language connecting abstract groups with permutations, geometric symmetry, linear representations, and counting. In finite group theory, actions on cosets produce normal subgroups and permutation representations, while orbit counting leads directly to conjugacy classes, the Sylow theorems, the Cauchy–Frobenius–Burnside formula, and local structure. [@isaacs2008]

Definition

For fixed \(g\), the map \(x\mapsto g\cdot x\) has inverse \(x\mapsto g^{-1}\cdot x\), so it is a permutation. Giving an action is therefore equivalent to giving a homomorphism

\[ \rho:G\longrightarrow\operatorname{Sym}(X), \]

called the permutation representation of the action.

A right action is written \(x\cdot g\) and satisfies \(x\cdot1=x\) and \((x\cdot g)\cdot h=x\cdot(gh)\). It becomes a left action by defining \(g*x=x\cdot g^{-1}\). Isaacs predominantly uses right actions; this article uses left actions for general statements and specifies the side in coset examples.

Kernel, faithfulness, and invariant subsets

The action is faithful if \(\ker\rho=1\); then \(G\) is isomorphic to a subgroup of \(\operatorname{Sym}(X)\). A subset \(Y\subseteq X\) is \(G\)-invariant if \(gY=Y\) for every \(g\in G\). Such a subset is necessarily a union of orbits.

Orbits and stabilizers

Distinct orbits are disjoint, so the orbits partition \(X\). The action is transitive precisely when \(X\) is one orbit.

If \(y=g\cdot x\), then \(G_y=gG_xg^{-1}\), so stabilizers of points in the same orbit are conjugate.

Four fundamental actions

The class equation

For conjugation on a finite group, the orbit decomposition gives

\[ |G|=|Z(G)|+\sum_i|G:C_G(x_i)|, \]

where one \(x_i\) is selected from each noncentral conjugacy class.

This result drives inductive arguments for finite \(p\)-groups, central series, and many constructions in Sylow theory.

Fixed-point counting for p-groups

If a finite \(p\)-group \(P\) acts on a finite set \(X\), every orbit has prime-power size. If \(X^P\) denotes the set of points fixed by all of \(P\), then

\[ |X|\equiv|X^P|\pmod p. \]

In particular, an action on a set whose size is not divisible by \(p\) has a fixed point. The Sylow containment theorem follows by letting a \(p\)-subgroup act on the cosets of a Sylow subgroup.

Cauchy–Frobenius–Burnside orbit count

For a finite action, write \(X^g=\{x:g\cdot x=x\}\).

This formula counts colorings, necklaces, graphs, and molecular configurations up to rotations or reflections.

Common properties of actions

  • Free: every point stabilizer is trivial.
  • Transitive: every two points lie in one orbit.
  • Regular: free and transitive; exactly one group element maps any specified point to another.
  • n-transitive: transitive on ordered \(n\)-tuples of distinct points.
  • Primitive: transitive with no nontrivial block system.

These notions are not interchangeable. A transitive action can have nontrivial stabilizers, and a faithful action need not be free.

Example: an equilateral triangle

The symmetry group \(D_6\cong S_3\) acts faithfully and transitively on the three vertices. A vertex stabilizer consists of the identity and the reflection through that vertex, so it has order two. Orbit–stabilizer gives

\[ 3=|S_3:C_2|. \]

The rotation subgroup \(C_3\) acts regularly.

Applications

  • Coset actions turn small-index subgroups into homomorphisms to \(S_n\) and can force normal subgroups.
  • Conjugation actions produce class equations, centralizers, normalizers, and fusion questions.
  • A linear action is the starting point of representation theory: it is a homomorphism \(G\to GL(V)\).
  • Actions on graphs, geometries, topological spaces, and algebraic objects encode symmetry.
  • The Sylow theorems, Frobenius groups, and permutation-group theory all use actions as their basic language.

The English and Chinese Wikipedia articles on group actions were used to cross-check terminology, examples, and scope. The finite-group proofs were independently organized along Isaacs. The cited revisions are attributed under CC BY-SA. [@wikipedia-group-action-zh][@wikipedia-group-action-en]