Problem

Write out the cycle decomposition of the eight permutations in \(S_4\) corresponding to the elements of \(D_8\) given by the action of \(D_8\) on the vertices of a square (where the vertices of the square are labelled as in Section 2).

Solution

Vertices 1 to 4 are given in the clockwise direction,

• \(v_1 = (1,1)\)
• \(v_2 = (1,-1)\)
• \(v_3 = (-1,-1)\)
• \(v_4 = (-1,1)\)

and the reflection is with respect to the straight line through \(v_1\) and \(v_3\). For the rest, let 1 to 4 denote \(v_1\) to \(v_4\). Using the verbose form where identity mapping is explicitly presented,

• \(1 = (1)(2)(3)(4)\)
• \(r = (1\ 2\ 3\ 4)\)
• \(r^2 = (1\ 3)(2\ 4)\)
• \(r^3 = (1\ 4\ 3\ 2)\)
• \(s = (2\ 4)(1)(3)\)
• \(sr = (1\ 4)(2\ 3)\)
• \(sr^2 = (1\ 3)(2)(4)\)
• \(sr^3 = (1\ 2)(3\ 4)\)

$$\tag*{$\blacksquare$}$$