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

Question

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.

Answer 1

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 $S_6$.