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Let $g ∈ S_n$ be a permutation. Describe a method for answering the following question: does there exist a permutation $f ∈ S_n$ such that $f ◦ f = g$?

I don't want to be spoonfed the answer, but can someone give me a direction? It is getting late and my brain is running on fumes, so I can't think of a point to start.

  • @Randall Oh I see, I am going to try to solve it on my own, with some directions, and then check that page to confirm. –  Mar 19 '19 at 03:13

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Start by taking a cycle decomposition of $g$. What needs to be true of the cycles?

Michael Biro
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  • They have to be disjoint, but where my brain is stuck is: if f is a single cycle, it cannot possibly be disjoint from itself –  Mar 19 '19 at 03:15
  • However ,if $f = c_1 c_2 ... c_k$ , then $f^2 = (c_1)^2 (c_2)^2 ... (c_k)^2$, because cycles commute, so g has to be a product of disjoint square cycles –  Mar 19 '19 at 03:26