# Example of a non-abelain finite group $G$ with $G/N$ abelian and infinite group $G$ with $G/N$ finite

Have not been able to think of a examples with the following properties:

1. Example of a non-abelian finite group $G$ with property that $G/N$ is abelian for every non-trivial normal subgroup $N$ of $G$.

2. Example of an infinite group $G$ with property that $G/N$ is finite for every non-trivial normal subgroup $N$ of $G$.

Any help would be greatly appreciated.

• For a nonabelian non-simple example of (2) take SL(3,Z). – Moishe Kohan Jan 10 '15 at 14:56

Hint: (1) $Q_8, S_3.$ (2) $\mathbb Z.$
• (1) I have never come accross $Q_8$ so dont worry about that one, but thanks for the $S_3$ example, how does $S_3$ have this property? I cant see how this is the case. (2) Totally understand this one, should have thought of that. – Sam Houston Jan 10 '15 at 14:56
• What is the order of $S_3/A_3$? – user4601931 Jan 10 '15 at 15:06
• @Dansmith: $Q_8$ stands for Quaternion group. See en.wikipedia.org/wiki/Quaternion_group Note that $|S_3| = 6.$ So for any non-trivial normal subgroup $N$ of $S_3,$ the possible orders of $S_3/N$ is $2, 3.$ These are primes and hence cyclic. – Krish Jan 10 '15 at 15:15