Permutation $ \pi$ has a signature $2^43^5$. Find number of permutation $\sigma$ such that $\sigma^4 = \pi$
Could you give me a clue ?
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Hint: Consider the primes $2, 3$ separately.
What is the fourth power of a $3$-cycle?
Now note that for the $2$-cycle (transposition) bit, if you square a transposition, you will get the identity. When you square this fourth power, you get the identity in the $2$ bit of your element. Can you think of a kind of permutation which would have that property? Can you find a permutation of that kind whose fourth power is a product of four transpositions?