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Let $n$ be an odd number. Let $C_n$ be the set of permutations $\pi$ of $[n]$ whose cycle representation has only one cycle. Let $\pi,\sigma\in C_n$. Prove that their composition $\pi\sigma$ has an odd number of cycles.

For example, we have $(123)(132)=(1)(2)(3)$ has $3$ cycles, while $(123)(123)=(132)$ has $1$ cycle. Without loss of generality, we can suppose that $\pi=(12\cdots n)$.

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I think I get the solution. Let $\alpha $ and $\beta $ be two $n$-cycles in ${S_n}$. Since $n$ is odd, $\alpha $ and $\beta $ are even and so their composition $\alpha \beta $ is even. Since $\alpha \beta $ is even, $\alpha \beta $ must has even number disjoint cycles that are odd permutation. Otherwise $\alpha \beta $ would be odd. By odd(even) cycle, I mean that a cycle that is odd(even) permutation. Let $a$, $b$ and $c$ be the number of odd cycles, even cycles that differ from idendity and idendities which are included in $\alpha \beta $, respectively. For example, let $x = \left( {12} \right)\left( {349} \right)\left( {578} \right)\left( 6 \right)$ in ${S_9}$, then $a = 1$, $b=2$ and $c=1$. Now, what I need to show is that $a+b+c$ is odd. By above observation, $a$ must be even. What about $b$ and $c$? Note that an odd(even) cycle consists of even(odd) numbers and note that idendity consists of only one number. Now, suppose $b$ is even. Then, $E + E + c = n$. Since $n$ is odd, $c$ is odd and so $a+b+c$ is odd. Now, suppose $b$ is odd. Then, since $E + O + c = n$ and $n$ is odd, $c$ is even and so $a+b+c$ is odd. Here, E is an even number and O is odd number.

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  • $\begingroup$ I think you are right. Thank you! $\endgroup$ – kwgl Apr 25 '14 at 22:34
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In general, it is not true. For example, let $\alpha = \left( {123} \right)$ and $\beta = \left( {456} \right)$ in ${S_7}$. Then, $\alpha \beta = \left( {123} \right)\left( {456} \right)$ has two cycles.

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    $\begingroup$ Maybe I did not state clearly. In $S_7$, the permutation $\alpha=(123)$ is in fact $(123)(4)(5)(6)(7)$. It has $5$ cycles, rather than "only one cycle". The same thing happens to the $\beta$ in your example. The resulting permutation $\alpha\beta$ has $3$ cycles: $(123)(456)(7)$. $\endgroup$ – kwgl Apr 18 '14 at 1:02

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