Cayley's Theorem on symmetric groups

Cayley’s The­o­rem states that ev­ery group \(G\) ap­pears as a cer­tain sub­group of the sym­met­ric group \(\mathrm{Sym}(G)\) on the un­der­ly­ing set of \(G\).

For­mal statement

Let \(G\) be a group. Then \(G\) is iso­mor­phic to a sub­group of \(\mathrm{Sym}(G)\).

Proof

Con­sider the left reg­u­lar ac­tion of \(G\) on \(G\): that is, the func­tion \(G \times G \to G\) given by \((g, h) \mapsto gh\). This in­duces a ho­mo­mor­phism \(\Phi: G \to \mathrm{Sym}(G)\) given by cur­ry­ing: \(g \mapsto (h \mapsto gh)\).

Now the fol­low­ing are equiv­a­lent:

  • \(g \in \mathrm{ker}(\Phi)\) the ker­nel of \(\Phi\)

  • \((h \mapsto gh)\) is the iden­tity map

  • \(gh = h\) for all \(h\)

  • \(g\) is the iden­tity of \(G\)

There­fore the ker­nel of the ho­mo­mor­phism is triv­ial, so it is in­jec­tive. It is there­fore bi­jec­tive onto its image, and hence an iso­mor­phism onto its image.

Since the image of a group un­der a ho­mo­mor­phism is a sub­group of the codomain of the ho­mo­mor­phism, we have shown that \(G\) is iso­mor­phic to a sub­group of \(\mathrm{Sym}(G)\).

Parents:

  • Symmetric group

    The sym­met­ric groups form the fun­da­men­tal link be­tween group the­ory and the no­tion of sym­me­try.