I'm trying to teach myself Cosets. Can anyone help with this question:
Let $G$:= $\Sigma_6$ and $x$ = $(123)(456)$ $\in$ $\Sigma_6$. Determine $C_G$($x$) by finding a set $S \subseteq G$ with $C_G$($x$) = $\langle$ $S$ $\rangle$.
Hint: Consider the subgroup $H$ $\leq$ $C_G$($x$) consisting of all elements $h \in$ $G$ with $(123)^h$ = $(123)$ and $(456)^h$ = $(456)$ and describe the right cosets of $H$ in $C_G$($x$).
Any help would be appreciated. Thanks.