When I read the textbook about semidirect products and split extensions, I feel like I'm lacking the intuition behind them and their relations. I was wondering if someone could briefly explain such relations to me.
Specifically, this is what I'm struggling with:
Let α : K → Aut(H) be a homomorphism. By the semidirect product of H and K with respect to α, written $H \rtimes_α K$, we mean the set H×K with the binary operation given by setting $(h_1, k_1) · (h_2, k_2) = (h_1 · α(k_1)(h_2), k_1k_2)$.
For a splitting, it says:
A section, or splitting, for f : G → K is a homomorphism s : K → G, such that f ◦ s is the identity map of K. A homomorphism f : G → K that admits a section is said to be a split surjection. An extension of H by K is called a split extension if f : G → K admits a section.
The section s is said to split f.
For an extension, it says:
Let H and K be groups. An extension of H by K consists of a group G containing H as a normal subgroup, together with a surjective homomorphism f : G → K with kernel H.
Also, in a proposition, it says that split extensions are semidirect products.
But I'm not sure how an extension can "fit" the definition of a semidirect product.
Also, I think I have trouble understanding the relationship between these two when it comes to classifying groups. For semidirect products, I understand that semidirect products are not unique, so classifying them means looking at the possibilities of what groups a semidirect product might represent up to isomorphism. Is that correct? But what do extensions have to do with this?
Best Answer
I tried to explain it to you by answering your concrete question (Nonabelian group of order $p^3$ and semidirect products) yesterday, but maybe it was too short. Let's give it another try:
A group $G$ is the semidirect product of two groups $H$ and $K$ ($G\cong H\rtimes_\alpha K$ for some $\alpha$) if and only if there is a short exact sequence of groups $$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0,$$ that splits.
How to do this?
Let's assume $G\cong H\rtimes_\alpha K$, then $H$ is normal in $G$ and we get and exact sequence $$0\rightarrow H\rightarrow G\rightarrow (G/H)\cong K\rightarrow 0$$ and this sequence splits via the inclusion $K\hookrightarrow G$.
So the other way around: Take an exact sequence $$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0,$$ that splits via $\beta\colon K\rightarrow G$. Define $\alpha\colon K\rightarrow Aut(H),k\mapsto C_{\beta(k)}|_H,$ where $C_g$ denotes the conjugation with $g\in G$ in the group $G$. This induces an automoprhism on $H$ if we restrict it to $H$, since $H$ is normal in $G$.
Of course I did a lot of identifications like $H\cong H\times\{0\}$ and I hope you can work this out.
So we have a correspondence between splitting short exact sequences and semidirekt products. Short exact sequences of the shape $$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$ are just extensions of $K$ with $H$ to $G$. So classification of semidirect products is equivalent to classify splitting short exact sequences or let's call it splitting extensions.