Write down the elements of the cyclic subgroups in S8 generated by α and by α^-1 respectively.

Question

Let S8 be the symmetric group of degree 8.

1 2 3 4 5 6 7 8
6 8 3 7 1 4 5 2

Write down the elements of the cyclic subgroups in S8 generated by α and by α^(−1) respectively.

So I found a^(-1) to be

12345678
58367142

and a in disjoint form is (16475)(28)

a^(-1) in disjoint form is (15746)(28)

what do I do now?

Answer 1

First of all any element and its inverse (in any group) generate the same subgroup, this easily follows from the fact that any subgroup which contains $\alpha$ must contain $\alpha^{-1}$ and vice versa. So you don't need to think of this as two different exercises.

Anyway, the group which is generated by one element is $\langle\alpha\rangle=\{\alpha^k:k\in\mathbb{Z}\}$. If the order of $\alpha$ is finite and equal to $n$ then it is equal to $\{\alpha^k: 0\leq k\leq n-1\}$. So just start multiplying $\alpha$ by itself until you get the identity. This way you will get the whole subgroup $\langle\alpha\rangle$.