Non abelian group with normal subgroup

Question

Normal Subgroups are subgroups where all left cosets are right cosets. For abelian groups, all subgroups are normal.

I want to discuss about a non-abelian group whose subgroups are all normal. Please give an example.

Can we give example of a finite non-abelian group with same property?

Answer 1

The quaternion group is a finite, nonabelian group where every subgroup is normal.

Answer 2

If $H$ is a finite subgroup of a group $G$ then it has only finitely many conjugates $k^{-1}Hk$ and these all have finite index in $G$ also. Taking the intersection of these subgroups, $N=\cap_{k\in G}k^{-1}Hk$ yields a normal subgroup which also has finite index.

This is a constructive method of creating normal subgroups. However, if $G$ is finite then the subgroup $N$ may be trivial. Note that if $G$ is infinite and simple then the above working implies that $G$ contains no subgroups of finite index.