Permutation Representation

A permutation representation of a group $G$ on a set $X$ is a group homomorphism

where $\mathrm{Sym}(X)$ denotes the group of all bijective maps $X\to X$ under composition. Giving such a homomorphism is equivalent to giving a group action via $g\cdot x := \rho(g)(x)$.

The kernel of $\rho$ is exactly the kernel of the action . In particular, $\rho$ is injective iff the action is faithful , and Cayley's theorem says every group has a faithful permutation representation (on itself by left multiplication).

Examples:

• (Left regular representation) $\rho(g)$ is the permutation $x\mapsto gx$ of the underlying set of $G$.
• (Action on cosets) For $H\le G$, the action on $G/H$ gives a homomorphism $G\to \mathrm{Sym}(G/H)$.
• (Conjugation) The conjugation action gives a homomorphism $G\to \mathrm{Sym}(G)$.