Editing Permutation group (section)

==Group actions==
{{main|Group action (mathematics)}} 
In the above example of the symmetry group of a square, the permutations "describe" the movement of the vertices of the square induced by the group of symmetries. It is common to say that these group elements are "acting" on the set of vertices of the square. This idea can be made precise by formally defining a '''group action'''.<ref name=Dixon96>{{harvnb|Dixon|Mortimer|1996|loc=p. 5}}</ref>

Let ''G'' be a [[Group (mathematics)|group]] and ''M'' a nonempty [[Set (mathematics)|set]]. An '''action''' of ''G'' on ''M'' is a function ''f'': ''G'' × ''M'' → ''M'' such that
* ''f''(1, ''x'') = ''x'', for all ''x'' in ''M'' (1 is the [[Identity element|identity]] (neutral) element of the group ''G''), and
* ''f''(''g'', ''f''(''h'', ''x'')) = ''f''(''gh'', ''x''), for all ''g'',''h'' in ''G'' and all ''x'' in ''M''.
This pair of conditions can also be expressed as saying that the action induces a group homomorphism from ''G'' into ''Sym''(''M'').<ref name=Dixon96 /> Any such homomorphism is called a ''(permutation) representation'' of ''G'' on ''M''.

For any permutation group, the action that sends (''g'', ''x'') → ''g''(''x'') is called the '''natural action''' of ''G'' on ''M''. This is the action that is assumed unless otherwise indicated.<ref name=Dixon96 /> In the example of the symmetry group of the square, the group's action on the set of vertices is the natural action. However, this group also induces an action on the set of four triangles in the square, which are: ''t''<sub>1</sub> = 234, ''t''<sub>2</sub> = 134, ''t''<sub>3</sub> = 124 and ''t''<sub>4</sub> = 123. It also acts on the two diagonals: ''d''<sub>1</sub> = 13 and ''d''<sub>2</sub> = 24.
{| class="wikitable"
|-
! Group element !! Action on triangles !! Action on diagonals
|-
| (1) || (1) || (1)
|-
| (1234) || (''t''<sub>1</sub> ''t''<sub>2</sub> ''t''<sub>3</sub> ''t''<sub>4</sub>) || (''d''<sub>1</sub> ''d''<sub>2</sub>)
|-
| (13)(24) || (''t''<sub>1</sub> ''t''<sub>3</sub>)(''t''<sub>2</sub> ''t''<sub>4</sub>) || (1)
|-
| (1432) || (''t''<sub>1</sub> ''t''<sub>4</sub> ''t''<sub>3</sub> ''t''<sub>2</sub>) || (''d''<sub>1</sub> ''d''<sub>2</sub>)
|-
| (12)(34) || (''t''<sub>1</sub> ''t''<sub>2</sub>)(''t''<sub>3</sub> ''t''<sub>4</sub>) || (''d''<sub>1</sub> ''d''<sub>2</sub>)
|-
| (14)(23) || (''t''<sub>1</sub> ''t''<sub>4</sub>)(''t''<sub>2</sub> ''t''<sub>3</sub>) || (''d''<sub>1</sub> ''d''<sub>2</sub>)
|-
| (13) || (''t''<sub>1</sub> ''t''<sub>3</sub>) || (1)
|-
| (24) || (''t''<sub>2</sub> ''t''<sub>4</sub>) || (1)
|}