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

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