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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'').

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.


PROOF OF THE THEOREM

  • ''G'' = ''G''; and by cancellation rules, that ''g''---''x'' = ''g''---''y'' if and only if ''x'' = ''y''. So multiplication by ''g'' acts as a Bijective function ''f''''g'' : ''G'' → ''G'', by defining ''f''''g''(''x'') = ''g''---''x''. Thus, ''f''''g'' is a permutation of ''G'', and so is a member of Sym(''G'').


  • ''x'' for all ''x'' in ''G''} is a subgroup of Sym(''G'') which is

    • isomorphic to ''G''. The fastest way to establish this is to consider the

  • x'' = ''x'' for all ''x'' in ''G'', and taking ''x'' to be the identity element ''e'' of ''G'' yields ''g'' = ''g''---''e'' = ''e''.


Thus ''G'' is isomorphic to the image of ''T'', which is the subgroup ''K'' considered earlier.

''T'' is sometimes called the ''regular representation of'' ''G''.


Alternate setting of proof

An alternate setting uses the language of Group Action s. We consider the group G as a G-set, which can be shown to have permutation representation, say \phi.

Firstly, suppose G=G/H with H=\{e\}. Then the group action is g.e by Classification Of G-orbits (also known as the orbit-stabilizer theorem).

Now, the representation is faithful if \phi is injective, that is, if the kernel of \phi is trivial. Suppose g ∈ ker \phi Then, g=g.e=\phi(g).e by the equivalence of the permutation representation and the group action. But since g ∈ ker \phi, \phi(g)=e and thus ker \phi is trivial. Then im \phi < G and thus the result follows by use of the First Isomorphism Theorem .


REMARKS ON THE REGULAR GROUP REPRESENTATION

The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left Coset of the subgroup generated by the element.


EXAMPLES OF THE REGULAR GROUP REPRESENTATION

Z2 = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12).

Z3 = {0,1,2} with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E.g. 1 + 1 = 2 corresponds to (123)(123)=(132).

Z4 = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).

The elements of Klein Four-group {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).

S3 ( Dihedral Group Of Order 6 ) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements: