Hw7 Problem 4

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Problem 4

Show that $A_{8}$ has an element of order $15$.

Solution

We know that the order of the product of two disjoint cycles is the least common multiple of the lengths of the cycles. Therefore, the product of a cycle of length $5$ and a cycle of length $3$ will have an order of $15$. For instance: $(1 \ 2 \ 3 \ 4 \ 5)(6 \ 7 \ 8)$ has an order of $15$.