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Find the largest positive integer $k$ such that $S_5 \times S_5$ has an element of order $k$.

I know by Lagrange that the order of any element of $S_5\times S_5$ divides $\left\lvert S_5\times S_5\right\rvert$. I assume that this is the way into the question but I'm not sure what to do from here.

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  • $\begingroup$ Start with this question. $\endgroup$ – Dietrich Burde May 9 '16 at 19:37
  • $\begingroup$ @DietrichBurde So is it 30? $\endgroup$ – Si.0788 May 9 '16 at 19:47
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Hint:

For an element $(a, b) \in S_5 \times S_5$, the order of that element is the LCM of the order of $a$ and the order of $b$. You know that the order of $a$ and the order of $b$ both have to divide the order of $S_5$, which is $120$ (but there are other limits, too, so this answer might help you find the different orders of elements in $S_5$). How can you maximize the LCM of their orders?

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  • $\begingroup$ So is the answer 30? $\endgroup$ – Si.0788 May 9 '16 at 19:50
  • $\begingroup$ @Si.0788 Yes, I'm pretty sure that's correct. Good job! $\endgroup$ – Noble Mushtak May 9 '16 at 20:21

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