Let $p$ be a prime and let $n$ be an integer such that $n \le p$.
a) Let $\alpha$ be a p-cycle in $S_n$. Prove that $\alpha^j$ is a p-cycle for $1 \le j \le p − 1$.
b) Assume that $2p \le n$. Let $\alpha$ and $\beta$ be disjoint p-cycles in $S_n$. Prove that $|\alpha^i\beta^j| = p$ for $1 \le i,\quad j \le p − 1$.
c) Let $\alpha$ and $\beta$ be as in part (b). Prove that if $\alpha^i\beta^j = \alpha^k \beta^l$ for integers i, j, k and $l$, then p divides i − k and j − $l$.
d) Prove that if $H$ is a subgroup of $S_7$ and $|H| = 9$, then $H$ is not cyclic.
e) Find an abelian subgroup $H$ of $A_7$ such that $|H| = 9$.
I'm particularly stuck on (a), (b), and (d). Any help would be appreciated.