Cayley's theorem

In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G.

Related Topics:
Group theory - Arthur Cayley - Group - Isomorphic - Subgroup - Symmetric group - Group action

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A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G).

Related Topics:
Permutation - Bijective - Function

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Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.

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