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 $G$.

Also, please explain why.

Any help would be greatly appreciated.

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

Hint: (1) $Q_8, S_3.$ (2) $\mathbb Z.$

  • $\begingroup$ (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. $\endgroup$ – Sam Houston Jan 10 '15 at 14:56
  • $\begingroup$ What is the order of $S_3/A_3$? $\endgroup$ – user4601931 Jan 10 '15 at 15:06
  • $\begingroup$ @dmdmdmdmdmd: it's two. $\endgroup$ – Krish Jan 10 '15 at 15:13
  • $\begingroup$ Right. So what do you know about groups of prime order? $\endgroup$ – user4601931 Jan 10 '15 at 15:14
  • $\begingroup$ @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. $\endgroup$ – Krish Jan 10 '15 at 15:15

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