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> Let $p$ be an odd prime and let $n$ be an integer such that $n \ge 3p$. Let $\alpha$ be a $2p$-cycle in $S\_n$.
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> a) For which values of $j \in \{ 1, . . . , 2p − 1 \}$ is $\alpha^j$ a $2p$-cycle?
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> b) If $1 \le j \le 2p−1$ and $\alpha^j$ is not a $2p$-cycle, what is the cycle decomposition of $\alpha^j$?
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> c) Let $\beta$ be a $p$-cycle in $S\_n$. Assume that $\alpha$ and $\beta$ are disjoint. Let 
>  $H = \{ \alpha^j\beta^i \mid 0 \le j \le 2p − 1, 0 \le i \le p − 1\}$. Prove that $H$ is an abelian subgroup of $S\_n$.
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> d) Let $\beta$ be as in part c). Suppose that $K$ is a subgroup of $S\_n$ that contains $\alpha$ and $\beta$. Prove that $K$ is not cyclic.
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for a) If I take $p = 3$, then shouldnt all elements 1,2,3,4,5 result in $\alpha^j$ be a 6 cycle?

Im not sure of my first part so im stuck for the rest of the problems, Im not expecting answers but I would really appreciate guidance as to which direction I should go on.

Thanks