Find the orders of the following permutation subgroups of $S_4$:
a) The subgroup generated by $(1,2), (3,4)$ and $(1,3)$.
b) The subgroup generated by $(1,2), (3,4)$ and $(1,3)(2,4)$.
I cannot come up with any general method to compute the orders of such permutation groups, except for listing all possibilities. But the problem is, when I did the listing, it would be quite tedious. Could anyone give me some help in this problem?
Best Answer
For part (a), the trick is to remember that $S_4$ is generated by the adjacent transpositions $(1,2)$, $(2,3)$, $(3,4)$. You already have the adjacent transpositions $(1,2)$ and $(3,4)$. You can get the last adjacent transposition as $(2,3)=(1,2)(1,3)(1,2)$.
For part (b), consider the square with vertices labeled by $$\begin{array}{c c}1&3\\4&2\end{array}$$ The permutation $(1,2)$ is a reflection across the southwest-northeast diagonal. The permutation $(3,4)$ is a reflection across the northwest-southeast diagonal. The permutation $(1,3)(2,4)$ is a reflection across the vertical line of symmetry. Thus, all three of your generators are symmetries of this square, so the subgroup generated by your generators must lie inside the symmetries of this square which is the dihedral group of order 8. It is not hard to convince yourself that you do get all of the dihedral group of order 8. For instance, the product $(1,2)(3,4)$ is also in your subgroup so your subgroup has at least 4 nonidentity elements. By Lagrange's theorem, the subgroup must be all of the dihedral group of order 8.