TITLE: Taking square root of cycle permutations
QUESTION [2 upvotes]: Let $\alpha$ and $\beta$ be permutation cycles of $\{1,2,\ldots,n\}$ such that $\alpha^2=\beta^2$ Can we conclude that $\alpha=\beta$, if
(a) $\alpha,\beta$ are odd?
(b) $\alpha,\beta$ are even?

We can write $\alpha=(a_1 a_2\ldots a_k), \beta=(b_1 b_2\ldots b_l)$, and we know that $\alpha^2=\beta^2$. I don't know how to continue to get more information on $\alpha,\beta$.

REPLY [3 votes]: Note that a cycle is odd if it has even length, and conversely.
Just experiment a little, and you will see that (a) is false.
For (b), notice that if $\alpha$ is a cycle of odd length, then $\alpha^2$ is also a cycle, of the same length as $\alpha$. (This is not true if $\alpha$ has even length.) Can you recover $\alpha$ from $\alpha^2$ ?
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