Example

Let be an equilateral triangle with vertices A, B, and C. The reflection in the line L through B and the midpoint of the edge AC induces a permutation of the three vertices:

A -> C

B -> B

C -> A.

If we name the three vertices as 1,2,3 for A, B, C, respectively, then we can describe the reflection by the permutation (1,3). A rotation through +120° is also a permutation of the three vertices. This rotation is described by the permutation (1,3,2). If we choose other names for the vertices, for example 1,3,2 for A, B, C, then the description of the reflection and the rotation change. The reflection is then for example described by (1,2) and the rotation by (1,2,3). This renumbering may be achieved by the permutation k = (2,3). Indeed, we see that

k (1,2)k^-1 = (1,3)

and

k (1,3,2)k^-1 = (1,2,3).

Conjugation is similar to basis transformation in linear algebra.