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Given an even permutation $\sigma,$ what more is required for it to be the square of some other permutation? Is it enough that there be an even number of transpositions in its cycle decomposition?

I looked at the case $\sigma=(12)(3456)$ and could not find a squareroot for that, or others I tried having an odd number of transpositions in their decompositions.

I would be interested in any specific counterexample having an odd number of transpositions, and also whether more is required for a permutation to be a square.

N. F. Taussig
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coffeemath
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    If you have the cycle decomposition of a permutation, try to figure out the cycle decomposition of its square. – user8268 May 01 '23 at 05:37
  • @user8268 I did that and it suggested what I conjectured about the cycle decomposition needing an even number of transpositions. However I wasn't myself sure, and also wondered about whether more conditions were needed to guarantee a permutation is a square. – coffeemath May 01 '23 at 06:13
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    your condition is necessary but not sufficient. There is a similar condition for other cycles of even length. – user8268 May 01 '23 at 06:19
  • You might try thinking about squares and square roots of single cycles. – paw88789 May 01 '23 at 12:20
  • To moderators: The following shows my question is a duplicate. Maybe it can be labelled as such in order for it not to remain open. [comments here already answer it.] https://math.stackexchange.com/questions/266569/how-to-find-the-square-root-of-a-permutation – coffeemath May 01 '23 at 16:08

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