Hw7 Problem 7

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### Problem 7

Find the maximum possible order for an element of $S_{10}$.

**Solution**

By HW7 Problem 3, when we write an element as the product of disjoint cycles, the order of the element will be the least common multiple of the lengths of the disjoint cycles. So, we need to find a sequence of numbers which sum to 10 and have the greatest possible least common multiple. The product of a disjoint 5, 3 and 2-cycle will have maximal order of 30.