A little confusion regarding Sylow $p$-subgroups generated by a $p$-cycle
Solution 1:
The equation $g^{h\hat{s}}=h^{-1}g^{\hat{s}}h$ is wrong, since $g^{h\hat{s}}=\hat{s}^{-1}h^{-1}gh\hat{s}$ with the $h$'s on the inside instead of the $\hat{s}$'s. To remove an $h$ from $s$ in this way it would need to be on the right instead of on the left. You could try cancelling the $g$'s and $h$'s from the right side of $s$ instead of the left, but then cancelling $g$'s would not work for a similar reason.