Exercise 1.3.4 (Orders of the elements of $S_3$, of $S_4$)

Proof.

(a)

Order of the elements of $S_3$: $$\begin{array}{cc}\hfill \sigma \hfill & \hfill |\sigma |\hfill \\ & \\ \hfill \hfill \\ \hfill ()\hfill & \hfill 1\hfill \\ \hfill (12)\hfill & \hfill 2\hfill \\ \hfill (13)\hfill & \hfill 2\hfill \\ \hfill (23)\hfill & \hfill 2\hfill \\ \hfill (123)\hfill & \hfill 3\hfill \\ \hfill (132)\hfill & \hfill 3\hfill \end{array}$$

(b)

Order of the elements of $S_4$: $$\begin{array}{cc}\hfill \sigma \hfill & \hfill |\sigma |\hfill \\ & \\ \hfill \hfill \\ \hfill ()\hfill & \hfill 1\hfill \\ \hfill (12)\hfill & \hfill 2\hfill \\ \hfill (1234)\hfill & \hfill 4\hfill \\ \hfill (13)(24)\hfill & \hfill 2\hfill \\ \hfill (134)\hfill & \hfill 3\hfill \\ \hfill (234)\hfill & \hfill 3\hfill \\ \hfill (1432)\hfill & \hfill 4\hfill \\ \hfill (1342)\hfill & \hfill 4\hfill \\ \hfill (1324)\hfill & \hfill 4\hfill \\ \hfill (1423)\hfill & \hfill 4\hfill \\ \hfill (1243)\hfill & \hfill 4\hfill \\ \hfill (243)\hfill & \hfill 3\hfill \\ \hfill (143)\hfill & \hfill 3\hfill \\ \hfill (14)(23)\hfill & \hfill 2\hfill \\ \hfill (142)\hfill & \hfill 3\hfill \\ \hfill (132)\hfill & \hfill 3\hfill \\ \hfill (13)\hfill & \hfill 2\hfill \\ \hfill (34)\hfill & \hfill 2\hfill \\ \hfill (24)\hfill & \hfill 2\hfill \\ \hfill (14)\hfill & \hfill 2\hfill \\ \hfill (23)\hfill & \hfill 2\hfill \\ \hfill (12)(34)\hfill & \hfill 2\hfill \\ \hfill (123)\hfill & \hfill 3\hfill \\ \hfill (124)\hfill & \hfill 3\hfill \\ \hfill \hfill \end{array}$$

We obtained the list of the elements of $S_4$ by

     sage: G = SymmetricGroup(4); G
     Symmetric group of order 4! as a permutation group
     sage: l = [sigma for sigma in G]; l
     
     [(),
      (1,2),
      (1,2,3,4),
      (1,3)(2,4),
      (1,3,4),
      (2,3,4),
      (1,4,3,2),
      (1,3,4,2),
      (1,3,2,4),
      (1,4,2,3),
      (1,2,4,3),
      (2,4,3),
      (1,4,3),
      (1,4)(2,3),
      (1,4,2),
      (1,3,2),
      (1,3),
      (3,4),
      (2,4),
      (1,4),
      (2,3),
      (1,2)(3,4),
      (1,2,3),
      (1,2,4)]