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In 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 {\displaystyle \operatorname {Sym} } whose elements are the permutations of the underlying set of G.Explicitly,
The homomorphism G → Sym {\displaystyle G\to \operatorname {Sym} } can also be understood as arising from the left translation action of G on the underlying set G.
When G is finite, Sym {\displaystyle \operatorname {Sym} } 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 S n {\displaystyle S_{n}}. But G might also be isomorphic to a subgroup of a smaller symmetric group, S m {\displaystyle S_{m}} for some m < n {\displaystyle m Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups".