Let X=Sym(n) and W=$\displaystyle C_X(t)=\{x\in X \mid xt=tx, \mbox{ where t is a transposition } (1 2)\}$. Let $\displaystyle Y=\{z \mid z \mbox { is a transposition } o(zs)=o(zt)=3 \}$. Then how do we prove that $\displaystyle X=\langle W, Y \rangle.$

Can use the facts $\displaystyle W \cong S_2 \times S_{n-2}$ and for $\displaystyle \alpha \in Aut(X)$, $\displaystyle \alpha(Y)=Y, \alpha(t)=t.$