If $f$ and $g$ are two permutations on $\{1,2,3,4,5,6,7,8,9,10\}$ such that none of them has a cycle of length $5$, is it possible for $f \circ g$ to have a cycle of length $5$?
I think this is possible because I have seen some composite permutations where the resulting permutation has a cycle of greater length than any of the two permutations in the composition. However I do not know how to approach this problem and prove it, I can just create an example where this is true but I want to know how to show that it is true without creating an example.
I do not think that the inverse permutation or the identity permutation would help me in this situation, any ideas of what I can do without telling me the answer?
Hint: $(a b c)(c d e) = (a b c d e)$