Does there exit a solution of equation $x^2=(1234)$ in $S_4$ Group?
In solution of this question the reason state that no because $x^2$ is odd permutations which is not possible for $S_4$ . So I try to make a general statement for this
“ Does $x^2$ equals to odd permutations have solutions in any symmetric group $S_n$.“
My answer is no because irrespective of what type of permutations $x$ is $x^2$ must be even Permutations . Also I have verified it for $S_4$ but still I am not completely certain about this statement.