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

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.

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

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?