Q:

Questions about products of $p$-cycles.

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.
Thanks.

A:

Hints:
a) If $\alpha$ is a $p$-cycle, what is the order of $\alpha$?  In particular, what are the generators of $\langle\alpha\rangle$?
b) The reason that they want $2p$ to be less than or equal to $n$ is so that the cycles can be disjoint.  In other words, they want you to show that the order of two disjoint $p$-cycles is $p$.  (We know this reformulation of the question is valid by part $(a)$.)   In general, can you express $|ab|$ in terms of $|a|$ and $|b|$, if $a$ and $b$ are disjoint cycles?
d) If $|H|=9$ and $H$ were cyclic with generator $x$, then $x$ would have order $9$.  Can there be a $9$-cycle in $S_7$?  What would have to happen for the product of disjoint cycles to have order $9$?  (This ties into part $b$.)

