Editing Permutation group (section)

== Examples ==

Consider the following set ''G''<sub>1</sub> of permutations of the set ''M'' = {1, 2, 3, 4}:

* ''e'' = (1)(2)(3)(4) = (1)
**This is the identity, the trivial permutation which fixes each element.
* ''a'' = (1 2)(3)(4) = (1 2)
**This permutation interchanges 1 and 2, and fixes 3 and 4.
* ''b'' = (1)(2)(3 4) = (3 4)
**Like the previous one, but exchanging 3 and 4, and fixing the others.
* ''ab'' = (1 2)(3 4)
**This permutation, which is the composition of the previous two, exchanges simultaneously 1 with 2, and 3 with 4.

''G''<sub>1</sub> forms a group, since ''aa'' = ''bb'' = ''e'', ''ba'' = ''ab'', and ''abab'' = ''e''. This permutation group is, as an [[abstract group]], the [[Klein group]] ''V''<sub>4</sub>.

As another example consider the [[Examples of groups#The symmetry group of a square: dihedral group of order 8|group of symmetries of a square]]. Let the vertices of a square be labeled 1, 2, 3 and 4 (counterclockwise around the square starting with 1 in the top left corner). The symmetries are determined by the images of the vertices, that can, in turn, be described by permutations. The rotation by 90° (counterclockwise) about the center of the square is described by the permutation (1234). The 180° and 270° rotations are given by (13)(24) and (1432), respectively. The reflection about the horizontal line through the center is given by (12)(34) and the corresponding vertical line reflection is (14)(23). The reflection about the 1,3−diagonal line is (24) and reflection about the 2,4−diagonal is (13). The only remaining symmetry is the identity (1)(2)(3)(4). This permutation group is known, as an abstract group, as the [[dihedral group]] of order 8.