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I'm trying to understand the semi-direct product of groups from either a categorical or a geometric perspective and failing miserably. The four things I'm hoping will fit into a coherent picture are:

  1. Semidirect product as left adjoint;
  2. Semidirect product in terms of torsors;
  3. Semidirect product as a homotopy quotient;
  4. Semidirect product in terms of subobjects of the product.

I'll try to explain my struggles with each point.

  1. I started by trying to find a universal property and stumbled upon the first description, but I feel it did not help me even understand why semidirect products are interesting. Let $\mathsf{Grp^2}$ be the arrow category of the category of groups, and let $\mathsf{Grp}\circlearrowleft$ be the category of groups acting on groups: an object is a group action of a group $G$ on a group $H$ that respects all the operations, an arrow is an equivariant one. Such actions are in bijection with group homomorphisms $G\to \mathsf{Aut}(H)$. If these hold, we'll say $H$ is a $G$-group. The right adjoint is the forgetful functor $\mathsf{Grp^2}\to \mathsf{Grp}\circlearrowleft$ taking a group homomorphism to action by conjugation. The left adjoint, if I understand correctly, takes a $G$-group $\alpha:G\times H\to H$ to the inclusion $G\to H\rtimes G$. But why even care about $G$-groups? What is the semidirect product actually doing?
  2. Seeking intuition I hoped torsors might be related. Now, I don't know anything about torsors from the algebraic-geometry standpoint - for me a $G$-torsor is just a uniquely transitive $G$-set. John Baez's Torsors Made Easy and the linked answer made it clear torsors are somehow related to semidirect products, but I still don't get how, and still don't have a geometric picture of the semidirect product.
    • What I understand of the answer is as follows: We start with an $H$-group $G$, and a $G$-torsor $S$. As a $G$-set, $S$ has $G$ as a group of autobijections. If we pick a unit in $S$ then we have a group isomorphism $G\cong S$ which makes $S$ into an $H$-group. This gives the set $S$ another group $H$ of autobijections. The semidirect product $G\rtimes H$ is the subgroup of $\mathsf{Sym}(S)$ generated by the subgroups $G,H\leq \mathsf{Sym}(S)$.
    • But what is the geometry of this story? As the question itself mentions, the group $H$ seems to be keeping track of "orientation", while $G$ is doing the "movement". Does the semidirect product reflect the geometry of a torsor? I just don't get it.
    • In Tao's Compactness and Contradiction I found a very useful page or two about torsors, with a brief discussion of the lamplighter group. The idea was to think of an element of $G$ as a ratio $\frac BA$ between elements in some convenient $G$-torsor of "states". This makes conjugation intuitive since $g\frac BA g^{-1}=\frac{gA}{gB}$ is the unique group element with initial state $gA$ and terminal state $gB$. I was very hopeful this intuitive approach might actually motivate the unpleasant multiplication of the semidirect product, but I haven't been able to make sense of that myself.
  3. Waiting until 1,2 make sense;
  4. Waiting until 1,2 make sense.
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  • $\begingroup$ You care about $G$-groups if you care about $G$-sets: it's the conceptual pullback of groups with $G$-sets. The semidirect product give some the universal group containing $H$ in which all the automorphisms given by some $G$-structure on $H$ become inner. This is somewhere in between viewpoints (1) and (3). $\endgroup$ – Kevin Carlson Nov 4 '16 at 22:45
  • $\begingroup$ @KevinCarlson thank you, but your comment is too terse for me to really understand. $\endgroup$ – Arrow Nov 4 '16 at 22:51
  • $\begingroup$ I just mean that, if you like groups, and you like group actions, then it seems to follow that you like actions of groups on groups. $\endgroup$ – Kevin Carlson Nov 4 '16 at 22:52
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    $\begingroup$ @KevinCarlson sure, if you put it that way. I think I'm confused because I don't understand what this has to do with geometry and the universal property seem a little useless. Also, I am interested in what you wrote about "the universal group containing $H$...", if you feel like explaining :) $\endgroup$ – Arrow Nov 4 '16 at 22:57
  • $\begingroup$ It's just a way $\endgroup$ – Kevin Carlson Nov 5 '16 at 19:07

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