Cayley’s theorem

Let G be a group, then G is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to a subgroupMathworldPlanetmathPlanetmath of the permutation groupMathworldPlanetmath SG

If G is finite and of order n, then G is isomorphic to a subgroup of the permutation group Sn

Furthermore, suppose H is a proper subgroupMathworldPlanetmath of G. Let X={Hg|gG} be the set of right cosetsMathworldPlanetmath in G. The map θ:GSX given by θ(x)(Hg)=Hgx is a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The kernel is the largest normal subgroupMathworldPlanetmath of H. We note that |SX|=[G:H]!. Consequently if |G| doesn’t divide [G:H]! then θ is not an isomorphism so H contains a non-trivial normal subgroup, namely the kernel of θ.

Title Cayley’s theorem
Canonical name CayleysTheorem
Date of creation 2013-03-22 12:23:13
Last modified on 2013-03-22 12:23:13
Owner vitriol (148)
Last modified by vitriol (148)
Numerical id 7
Author vitriol (148)
Entry type Theorem
Classification msc 20B35
Related topic CayleysTheoremForSemigroups