Cayley's theorem
that is, the equivariant maps from this formula_128-set to itself are in bijection with formula_128. But it is easy to see that (1) these maps form a group under composition, which is a subgroup of formula_145, and (2) the function which gives the bijection is a group homomorphism. (Going in the reverse direction, it associates to every formula_28 in formula_128 the equivariant map of right-multiplication by formula_28.) Thus formula_128 is isomorphic to a subgroup of formula_145, which is the statement of Cayley's theorem.