# Thread: Question in group theory.

1. ## Question in group theory.

Hi all,
A student I'm helping gave me a question.

Q: Give an example of a group G with a normal subgroup N such that N and G/N are abelian, but G is not abelian.

I'm fairly sure there is a dihedral group with this property but i don't have time to find an explicit example.

Can someone help please?

Side note: If we weaken the conditions slightly and replace abelian by solvable no example exists.

2. Originally Posted by whipflip15 Hi all,
A student I'm helping gave me a question.

Q: Give an example of a group G with a normal subgroup N such that N and G/N are abelian, but G is not abelian.

I'm fairly sure there is a dihedral group with this property but i don't have time to find an explicit example.

Can someone help please?

Side note: If we weaken the conditions slightly and replace abelian by solvable no example exists.
every dihedral group $\displaystyle D_n, \ n \geq 3,$ satisfies the condition. because $\displaystyle D_n=<a,b: \ a^2=b^n=1, \ ab=b^{-1}a>,$ and so you just choose $\displaystyle N=<b>.$

#### Search Tags

group, question, theory 