Let $\alpha,\beta\in S_6$ be given by:
$$\alpha = \begin{pmatrix}1&2&3&4&5&6\\3&5&1&2&6&4\end{pmatrix},\beta = \begin{pmatrix}1&2&3&4&5&6\\6&1&5&3&4&2\end{pmatrix}$$
(a) Find $\alpha^{-1}$ and $\beta^{-1}$
(b) Find permutations $\gamma,\delta\in S_6$ such that $\gamma\circ \alpha = \beta$ and $\alpha \circ \delta = \beta$
(c) Find the least positive integer $m$ such that $\alpha^m = \iota$, where $\iota$ denotes the identity permutation of $X_6$.
(d) Find a permutation $\pi\in S_6$ such that $\pi^2 = \beta$
(e) Does there exist a permutation $\pi\in S_6$ such that $\pi^2 = \alpha$? Either find $\pi$ or explain why such a permutation does not exist.
although I found a permutation (d)but it is not a good way to find a such permutation and how to explain (e) it can't be found above my way
