§ Internal versus External semidirect products

Say we have an inner semidirect product. This means we have subgroups N,KN, K such that NK=GNK = G, NN normal in GG and NK={e}N \cap K = \{ e \}. Given such conditions, we can realize NN and KK as a semidirect product, where the action of KK on NN is given by conjugation in GG. So, concretely, let's think of NN (as an abstract group) and KK (as an abstract group) with KK acting on NN (by conjugation inside GG). We write the action of kk on nn as nkknk1n^k \equiv knk^{-1}. We then have a homomorphism ϕ:NKG\phi: N \ltimes K \rightarrow G given by ϕ((n,k))=nk\phi((n, k)) = nk. To check this is well-defined, let's take s,sNKs, s' \in N \ltimes K, with s(n,k)s \equiv (n, k) and s(n,k)s' \equiv (n', k'). Then we get:
ϕ(ss)==ϕ((n,k)(n,k))definition of semidirect product via conjugation:=ϕ((nnk,kk))definition of ϕ:=nnkkkdefinition of nk=knk1:=nknk1kk=nknk=ϕ(s)ϕ(s) \begin{aligned} &\phi(ss') = \\ &=\phi((n, k) \cdot (n', k')) \\ &\text{definition of semidirect product via conjugation:} \\ &= \phi((n {n'}^k, kk')) \\ &\text{definition of $\phi$:} \\ &= n n'^{k} kk' \\ &\text{definition of $n'^k = k n' k^{-1}$:} \\ &= n k n'k^{-1} k k' \\ &= n k n' k' \\ &= \phi(s) \phi(s') \end{aligned}
So, ϕ\phi really is a homomorphism from the external description (given in terms of the conjugation) and the internal description (given in terms of the multiplication). We can also go the other direction, to start from the internal definition and get to the conjugation. Let gnkg \equiv nk and gnkg' \equiv n'k'. We want to multiply them, and show that the multiplication gives us some other term of the form NKNK:
gg=(nk)(nk)=nknk=insert k1k=nknk1kk=n(knk1)kk=N is normal, so knk1 is some other element nN:=nnkk=NK \begin{aligned} gg' \\ &= (n k) (n' k') \\ &= n k n' k' \\ &= \text{insert $k^{-1}k$: } \\ &= n k n' k^{-1} k k' \\ &= n (k n' k^{-1}) k k' \\ &= \text{$N$ is normal, so $k n' k^{-1}$ is some other element $n'' \in N$:} \\ &= n n'' k k' \\ &= N K \end{aligned}
So, the collection of elements of the form NKNK in GG is closed. We can check that the other properties hold as well.