Let $(G,.)$ and $(H,.)$ be two groups and $\phi:G \to \text{Aut}(N)$ be a group homomorphism. On the set $N \times G$ define a binary operation by: $(m,g)(n,h)=(m (\phi(g)(n)),gh)$ and we donote the set $N \times G$ with this binary operation as $N \rtimes G.$ I have shown that $N \rtimes G$ is a group.
I have to show that$\require{AMScd}$ \begin{CD}N @>{i}>> N \rtimes G @>{\pi} >> G \\\end{CD} is an extension of $G$ by $N,$ where $i: N \to N \rtimes G$ defined by $n \mapsto (n,e_{G})$ and $\pi: N \rtimes G \to G$ is defined by $(n,g) \mapsto g.$ What I exactly have to show ?
Clearly $i$ is an injection and $\pi$ is a surjection. Any help will be appreciated. Thank you.
All you have to show is that
(1) Ker$(i)=1$
(2) Ker$(\pi)$ is the image of $N$ under $i$
(3) $\pi $ is surjective
As you have said these are relatively obvious results