Editing Permutation group (section)

==Cayley's theorem==
{{main|Cayley's theorem}}

Any group ''G'' can act on itself (the elements of the group being thought of as the set ''M'') in many ways. In particular, there is a  [[regular group action|regular action]] given by (left) multiplication in the group. That is, ''f''(''g'', ''x'') = ''gx'' for all ''g'' and ''x'' in ''G''. For each fixed ''g'', the function ''f''<sub>''g''</sub>(''x'') = ''gx'' is a bijection on ''G'' and therefore a permutation of the set of elements of ''G''. Each element of ''G'' can be thought of as a permutation in this way and so ''G'' is isomorphic to a permutation group; this is the content of [[Cayley's theorem]].

For example, consider the group ''G''<sub>1</sub> acting on the set {1, 2, 3, 4} given above. Let the elements of this group be denoted by ''e'', ''a'', ''b'' and ''c'' = ''ab'' = ''ba''. The action of ''G''<sub>1</sub> on itself described in Cayley's theorem gives the following permutation representation:
:''f''<sub>''e''</sub> ↦ (''e'')(''a'')(''b'')(''c'')
:''f''<sub>''a''</sub> ↦ (''ea'')(''bc'')
:''f''<sub>''b''</sub> ↦ (''eb'')(''ac'')
:''f''<sub>''c''</sub> ↦ (''ec'')(''ab'').