## how to solve ..

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 $\alpha \in Aut(X)$, $\alpha(Y)=Y, \alpha(t)=t.$