# Center of Simple Abelian Group and Simple Nonabelian Group

center of simple abelian group and nonabelian

From definition of simple, it must have no proper non trivial normal subgroup.

In Abelian group case since Abelian group always normal that means the center must be a whole group only.

Since simple group's condition does not allow no proper non trivial normal subgroup.

But How about non abelian group case ? Is it {e} only ? since simple group's condition restrict nontrivial normal subgroup. Or it's does not exist?

Let $G$ be a non-Abelian simple group $\implies$ it does not have any proper normal subgroup. But $Z(G)$ is always a normal subgroup . Here $G$ is non-abelian so $Z(G)$ can not be the group $G$ itself. Therefore $Z(G) =\{e\}$.