1a) Prove that for any two 3-cycles $\displaystyle \sigma_1, \sigma_2 \in S_n (n>4)$ there exists a $\displaystyle \tau \in A_n $such that $\displaystyle \tau\sigma_1\tau^{-1}=\sigma_2$

1b) Let $\displaystyle Z(S_n) $ be the center of $\displaystyle S_n $. Prove that $\displaystyle Z(S_n)=Z(A_n)=\{e\}$ for all $\displaystyle n>3 $ and $\displaystyle Z(S_3)=\{e\}$

1c) Prove that the only nontrivial normal subgroup of $\displaystyle S_n (n>4)$ is $\displaystyle A_n$

1d) Give a nontrivial normal subgroup $\displaystyle N$ of $\displaystyle S_4$ such that $\displaystyle N \ne A_4$

can anyone help? thanks in advance