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Cayley's theorem
Representation of groups by permutations
Cayley's theorem
In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group Sym(G) whose elements are the permutations of the underlying set of G. Explicitly, for each g∈G, the left-multiplication-by-g map ℓg:G→G sending each element x to gx is a permutation of G, and the map G→Sym(G) sending each element g to ℓg is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of Sym(G). The homomorphism G→Sym(G) can also be understood as arising from the left translation action of G on the underlying set G. When G is finite, Sym(G) is finite too. The proof of Cayley's theorem in this case shows that if G is a finite group of order n, then G is isomorphic to a subgroup of the standard symmetric group Sn. But G might also be isomorphic to a subgroup of a smaller symmetric group, Sm for some m...
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