# Example of group with normal subgroup $N\ne\{e\}$ such that $N \cap Z(G)= \{ e\}$ and $G$ \ $N$ contains an element of order more than $2$

(i) Give example of a group (if exists ) which has a normal subgroup $N\ne\{e\}$ such that

$N \cap Z(G)= \{ e\}$ and $G$ \ $N$ contains an element of order more than $2$

(ii) Give example of a group (if exists ) such that for every normal subgroup $N\ne\{e\}$ of $G$ ,

$N \cap Z(G)= \{ e\}$ and $G$ \ $N$ contains an element of order more than $2$

Let $G=S_3\times S_3$ then $Z(G)=e\times e$ and take $N=A_3\times e$ then you are done.
• Answer of $i)$. – mesel Sep 25 '14 at 14:11