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If I have the group $S_{9}$ and $\sigma, \tau \in S_{9}$ where $\vert \sigma \vert = 5$ and $\vert \tau \vert = 6$ is it then possible to have $\vert \sigma \tau \vert = 30$ and $\vert \sigma \tau \vert = 12$ why/why not?

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    No element of $S_9$ has order $30$ (think about cycle types).2012-03-31
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    Not possible because $30\ne 12$...2012-04-01
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    @lhf, I took it to be two separate questions, one about order 30, another about order 12. But, who knows?2012-04-01

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$(12345)(234657)=(1365)(247)$.