The question I have is: Let $S_3 = [1,2,3]$. Give an example of two permutations $\alpha, \beta$ in $S_3$ for which $(\alpha\beta)^2 \neq \alpha^2\beta^2$.
This is my answer: Let $\alpha = (123), \beta=(13)$. Therefore, $(\alpha\beta)^2 = (132)^2$ and $\alpha^2\beta^2 = (12)^2(13)^2 = 1$.
Does this make sense? I'm really struggling with understanding the mapping of permutation groups. Thank you.