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Find an element $a \in A_6$ that is not of the the form $b^2$ for any $b\in S_6$.

My professor gave the hint that we should consider the disjoint cycles of $A_6$.

So elements in $A_6$ will have either 0 or 2 disjoint two cycles but I'm not sure how i should use this to go about finding $a$.

Any guidance would be appreciated.

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First step is to find all possible cycle structures for elements of $A_6$;

  1. $5$-cycle,

  2. $4$-cycle times $2$-cycle,

  3. $3$-cycle,

  4. product of two $3$-cycles,

  5. product of two $2$-cycles,

  6. identity element.

Second step is to think about which of these are, and which aren't, squares in $S_6$.

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