Cayley's theorem
Representation of groups by permutations / From Wikipedia, the free encyclopedia
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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.[1] More specifically, G is isomorphic to a subgroup of the symmetric group whose elements are the permutations of the underlying set of G. Explicitly,
- for each , the left-multiplication-by-g map sending each element x to gx is a permutation of G, and
- the map sending each element g to is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of .
The homomorphism can also be understood as arising from the left translation action of G on the underlying set G.[2]
When G is finite, 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 . But G might also be isomorphic to a subgroup of a smaller symmetric group, for some ; for instance, the order 6 group is not only isomorphic to a subgroup of , but also (trivially) isomorphic to a subgroup of .[3] The problem of finding the minimal-order symmetric group into which a given group G embeds is rather difficult.[4][5]
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".[6]
When G is infinite, is infinite, but Cayley's theorem still applies.