Find an element $\alpha\in A_6$... Find an element $\alpha$ that belongs to $A_6$ that is not of the form $\beta^2$ for any $\beta$ that belongs to $S_6$. 
I understand how it may be useful to consider the disjoint cycle forms of elements in $A_6$, but don't really understand how to proceed. Thanks for any help
 A: The different pairwise disjoint cycle decompositions within $\,A_6\,$ are:
-- Two $\,2-$cycles (transpositions), like $\,(12)(34)\;,\;(24)(35)\,$ , etc.
-- One two cycle and one four cycle, like $\,(12)(3456)\,$
-- Three cycles: $\,(123)\;,\;(246)\,$ , etc.
-- Two three cycles: $\,(123)(456)\,$ , etc.
-- Five cycles: $\,(12345)\,$ , etc.
Now: $\,(12)(34)=(1324)^2\,$ , so the first class above doesn't work, and clearly three and five cycles are out as well, as squares of odd-length cycles are again a cylce of the same length.
We're left only with the second and/or the fourth options...can you take it from here?
A: Convince yourself that any $n$-cycle an be written as a product of $n-1$ transpositions, and hence any permutation with cycle type $(\lambda_1,\cdots,\lambda_n)$ has sign congruent to $\sum_i(\lambda_i-1)$ mod $2$.
Next, convince yourself that if $\sigma$ is an $n$-cycle, then $\sigma^2$ is also an $n$-cycle if $n$ is odd, and otherwise is a product two disjoint $n/2$-cycles if $n$ is even. Hence, the cycle type of $\sigma^2$ may be determined by keeping the odd-length cycles and splitting the even-length cycles into two halves.
Third, check that the cycles types possible in $S_6$ (i.e. integer partitions of $6$) are:


*

*(6), (5,1), (4,2), (4,1,1), (3,3), (3,2,1), (3,1,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)


and that, upon squaring, the cycle types of squares in $S_6$ are:


*

*(3,3), (5,1), (2,2,1,1), (3,1,1,1), (1,1,1,1,1,1),


and that, the cycle types among the above of even parity are:


*

*(3,3), (5,1), (2,2,1,1), (3,1,1,1), (1,1,1,1,1,1),


and that, the single cycle type of $S_6$ with even parity not listed above is (4,2). Thus, the elements of $A_6\setminus S_6^2$ are precisely the permutations of cycle type (4,2).
