Question in group theory.

whipflip15 · #1 $Old$ December 4th, 2008, 12:44 AM

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.

NonCommAlg · #2 $Old$ December 4th, 2008, 01:05 AM

Quote:

Originally Posted by whipflip15 $View Post$

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 $D_n, \ n \geq 3,$ satisfies the condition. because $D_n=<a,b: \ a^2=b^n=1, \ ab=b^{-1}a>,$ and so you just choose $N=<b>.$