Hi people!


This my first question, here. I don't sure if it has a trivial answer, or not.


Let G a group, N normal subgroup in G. In which cases there is a subgroup in G isomorphic to G/N?


TIA



Given a normal subgroup N⊆G, when does G contain a subgroup isomorphic to G/N?
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 $N \rtimes G/N$.


