Exact short sequences and semidirect products

Question

In several sources I have read that a group is semidirect product $G = N \rtimes_\phi G$ iff there is a short exact sequence related, of the form $0 \rightarrow N \rightarrow \ G \rightarrow K \rightarrow 0$ (I'm not sure is this is exactly as it is). I am aware that some questions regarding this topic have been previously asked, but I wanted to know which homomorphisms are those who take part in that short exact sequence. The only thing I know is that $Im(f_i) = Ker(f_{i+1})$, for $f_i$ homomorphisms that "make" that exact sequence.

Answer 1

Given a group $G$ any normal subgroup $N\subseteq G$ is part of a short exact sequence $$0 \rightarrow N \rightarrow G \rightarrow G/N \rightarrow 0\tag{1}$$ where the first map is the inclusion and the second map is the quotient map.

Given a normal subgroup $N\subseteq G$ and an arbitrary subgroup $H$ we can form the product of subgroups $N\cdot H = \{nh \mid n\in N, h\in H\}$. If $N\cap H = \{e\}$ this is an inner semidirect product and one can show that there is a canonical isomorphism $G/N \cong H$. Thus the above short exact sequence can be modified to be $$0\rightarrow N \rightarrow G \rightarrow H \rightarrow 0\tag{2}$$ but we have the additional information that $H \subseteq G$. As it turns out, the composite $$H \overset{\subseteq}\rightarrow G \rightarrow G/N \overset{\cong}\rightarrow H$$ is the identity, making the sequence (2) right-split exact, with section given by the inclusion $H\subseteq G$.

Now in general, given groups $N$ and $K$ and a homomorphism $\varphi:K \rightarrow \operatorname{Aut}(N)$ one can construct a new group called the outer semidirect product $N \rtimes_\varphi K$. As it turns out this fits into a short exact sequence $$0 \rightarrow N \rightarrow N\rtimes_\varphi K \rightarrow K \rightarrow 0$$ where as in (2) the right morphism is given by the composite $$N\rtimes_\varphi K/ N \rightarrow (N\rtimes_\varphi K)/N \cong K$$ and which is right split with section given by the inclusion $K\subseteq N \rtimes_\varphi K$. In fact this gives a characterization of outer semidirect products: A short exact sequence $$0 \rightarrow N \rightarrow G \rightarrow K \rightarrow 0$$ is right-split, if and only if there is an isomorphism $G \cong N\rtimes_\varphi K$, or more precisely, if and only if there is an isomorphism of right split short exact sequences $$\begin{array}{rcccl} 0 \rightarrow N&\rightarrow &G&\rightarrow &K\rightarrow 0\\ ||&&\downarrow\cong&&||\\ 0 \rightarrow N&\rightarrow &N\rtimes_\varphi K&\rightarrow &K\rightarrow 0 \end{array}$$