A group action is a way for an abstract group to act as symmetries of a set. It is the bridge from group structure to counting, geometry, conjugacy, and representation theory.
Actions as homomorphisms into symmetric groups
Definition. A left action of \(G\) on a set \(X\) is a map \(G\times X\to X\), \((g,x)\mapsto g\cdot x\), such that \(e\cdot x=x\) and \((gh)\cdot x=g\cdot(h\cdot x)\). Equivalently, it is a homomorphism \(G\to\operatorname{Sym}(X)\).
Example. \(S_n\) acts on \(\{1,\ldots,n\}\). A group \(G\) acts on itself by left multiplication, and also by conjugation \(g\cdot x=gxg^{-1}\).
Orbits and stabilizers
Definition. The orbit and stabilizer of \(x\in X\) are \[ Gx=\{g\cdot x:g\in G\}, \qquad G_x=\{g\in G:g\cdot x=x\}. \]
Proposition. The orbits partition \(X\), and \(G_x\le G\) for every \(x\in X\).
Proof. The relation \(x\sim y\) if \(y=g\cdot x\) is reflexive, symmetric, and transitive, so its equivalence classes are the orbits. The stabilizer subgroup test follows from \((gh^{-1})\cdot x=g\cdot(h^{-1}\cdot x)=x\) when \(g,h\) fix \(x\).
Orbit-stabilizer theorem
Theorem. If \(G\) is finite and acts on \(X\), then for every \(x\in X\), \[ |Gx|=[G:G_x], \qquad |G|=|Gx|\,|G_x|. \]
Proof. The map \(G/G_x\to Gx\), \(gG_x\mapsto g\cdot x\), is well-defined and bijective. Well-definedness and injectivity both use exactly the condition that elements differ by something stabilizing \(x\).
Conjugation and the class equation
Theorem. If \(G\) is finite, conjugation gives the class equation \[ |G|=|Z(G)|+\sum_i [G:C_G(x_i)], \] where the sum runs over representatives of noncentral conjugacy classes and \(C_G(x_i)\) is the centralizer of \(x_i\).
Corollary. Every finite \(p\)-group has nontrivial center.
Proof. In a \(p\)-group, every noncentral conjugacy class has size divisible by \(p\). The class equation gives \(|Z(G)|\equiv |G|\equiv0\pmod p\), so the center has at least \(p\) elements.
Burnside counting
Theorem. If a finite group \(G\) acts on a finite set \(X\), then the number of orbits is \[ |X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|, \] where \(X^g=\{x\in X:g\cdot x=x\}\).
Example. The rotations of a square act on colorings of its vertices with two colors. Burnside counting averages the number of colorings fixed by rotations of angles \(0,90,180,270\) degrees, giving \((16+2+4+2)/4=6\) rotational colorings.