Given a normal subgroup N⊆G, when does G contain a subgroup isomorphic to G/N? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:14:38Z http://mathoverflow.net/feeds/question/26025 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26025/given-a-normal-subgroup-ng-when-does-g-contain-a-subgroup-isomorphic-to-g-n Given a normal subgroup N⊆G, when does G contain a subgroup isomorphic to G/N? ajlopez 2010-05-26T14:55:46Z 2010-11-05T13:24:33Z <p>Hi people!</p> <p>This my first question, here. I don't sure if it has a trivial answer, or not.</p> <p>Let G a group, N normal subgroup in G. In which cases there is a subgroup in G isomorphic to G/N?</p> <p>TIA</p> http://mathoverflow.net/questions/26025/given-a-normal-subgroup-ng-when-does-g-contain-a-subgroup-isomorphic-to-g-n/26038#26038 Answer by Simon Rose for Given a normal subgroup N⊆G, when does G contain a subgroup isomorphic to G/N? Simon Rose 2010-05-26T16:31:58Z 2010-05-26T16:31:58Z <p>Assuming you're looking at the case where the isomorphism is induced by the quotient $G \to G/N$ (as per George McNinch's comment), then this should be if and only if the sequence $$0 \to N \to G \to G/N \to 0$$ splits. i.e. there is a section $\sigma : G/N \to G$. This is then seen to be equivalent to $G$ being isomorphic to the semidirect product $N \rtimes G/N$.</p> http://mathoverflow.net/questions/26025/given-a-normal-subgroup-ng-when-does-g-contain-a-subgroup-isomorphic-to-g-n/26634#26634 Answer by Tilemachos Vassias for Given a normal subgroup N⊆G, when does G contain a subgroup isomorphic to G/N? Tilemachos Vassias 2010-05-31T22:56:23Z 2010-05-31T22:56:23Z <p>You have to be careful! Of course in split extensions it is trivial that $G/N$ is isomorphic to a subgroup of $G$. On the other hand there are examples of extensions $$1\rightarrow N\rightarrow G\rightarrow G/N\rightarrow 1$$ that are not split but nevertheless there is a subgroup $H\le G$ with $H\cong G/N$.</p> <p>An example would be the quaternion group that cannot be written as a nontrivial extension. But it contains a normal subgroup of index 2 and a subgroup of order 2.</p> <p>Unfortunatly I don't see a solution to your problem in general.</p> http://mathoverflow.net/questions/26025/given-a-normal-subgroup-ng-when-does-g-contain-a-subgroup-isomorphic-to-g-n/33765#33765 Answer by Jimmy for Given a normal subgroup N⊆G, when does G contain a subgroup isomorphic to G/N? Jimmy 2010-07-29T08:12:32Z 2010-07-29T08:12:32Z <p>A good theorem related to your problem is Schur-Zassenhaus theorem. It states that when the normal subgroup N is a Hall subgroup, namely the order of N and the index of N are coprime, then there exists a complement of N, that is a subgroup H s.t. G=NH and N\cap H={identity}. So H is isomorphic to G/N.</p>