# Cayley’s theorem

Let $G$ be a group, then $G$ is isomorphic to a subgroup of the permutation group $S_{G}$

If $G$ is finite and of order $n$, then $G$ is isomorphic to a subgroup of the permutation group $S_{n}$

Furthermore, suppose $H$ is a proper subgroup of $G$. Let $X=\{Hg|g\in G\}$ be the set of right cosets in $G$. The map $\theta:G\to S_{X}$ given by $\theta(x)(Hg)=Hgx$ is a homomorphism. The kernel is the largest normal subgroup of $H$. We note that $|S_{X}|=[G:H]!$. Consequently if $|G|$ doesn’t divide $[G:H]!$ then $\theta$ is not an isomorphism so $H$ contains a non-trivial normal subgroup, namely the kernel of $\theta$.

Title Cayley’s theorem CayleysTheorem 2013-03-22 12:23:13 2013-03-22 12:23:13 vitriol (148) vitriol (148) 7 vitriol (148) Theorem msc 20B35 CayleysTheoremForSemigroups