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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.

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Typo, want $n\ge p$ I think. –  André Nicolas Feb 22 '13 at 17:33
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What have you tried? Also, did you mean $p\leq n$? –  Tobias Kildetoft Feb 22 '13 at 17:33
    
Some induction may probably help here:for any cycle $\,\alpha\,$ of odd length, $\,\alpha^k\,$ is again a cycle of the same length, for any $\,k\in\Bbb N\,$ –  DonAntonio Feb 22 '13 at 19:38
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I object to whomever is downvoting. This guy's only asking for hints, not answers, and he has done $c$ and $e$ and attempted the others, so that shows research effort. –  Alexander Gruber Feb 22 '13 at 21:04
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1 Answer 1

up vote 2 down vote accepted

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 $b$.)

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