The semidirect product as a deformation of the direct product

Question

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?

Answer 1

This was supposed to be a comment but it got too long.

It is misleading to think of the semidirect product with $\varphi = \text{id}$ as the categorical product. As I explain in this answer, the semidirect product is a (higher) colimit rather than a limit, and in particular its (higher) universal property involves maps out of it rather than maps into it. One way to state this universal property when $\varphi = \text{id}$ is that $G \times H$ is the commutative coproduct of the groups $G$ and $H$: that is, it is the universal group equipped with maps from $G$ to $H$ whose images commute.

This is more visible once we generalize the semidirect product to other contexts. For example, if $G$ is a group acting on a $k$-algebra $R$, then there is a notion of crossed product algebra

$$R \rtimes_{\varphi} G \cong R \langle G \rangle / (grg^{-1} = \varphi(g) r)$$

which is a version of the semidirect product. Here by $R \langle G \rangle$ I mean a "noncommutative group algebra" constructed by starting from $R$ and adjoining noncommuting formal variables, one for each element of $G$, subject to the relations in $G$. When $\varphi = \text{id}$ this reproduces the tensor product $R \otimes_k k[G]$, which again is the commutative coproduct of algebras (and in particular is neither the product nor the coproduct).

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Having said that let me try to answer the original question. From the deformation perspective, there's no reason to restrict our attention to semidirect products: more generally we can consider arbitrary short exact sequences

$$1 \to N \to X \to G \to 1$$

as deformations of the trivial short exact sequence

$$1 \to N \to N \times H \to H \to 1$$

and ask if we can come up with a cohomology theory that describes this. The answer is yes. The thing we want to compute, which classifies the possible choices of $X$, is the nonabelian cohomology

$$H^1(G, \underline{\text{Aut}}(N)) \cong [BG, B \underline{\text{Aut}}(N)]$$

of $G$ with coefficients in the automorphism 2-group of $N$ (here $B$ denotes taking the classifying space). This is a gadget which tracks not only the automorphisms of $N$ but also the auto-natural transformations among those automorphisms, thinking of $N$ as a one-object category and automorphisms of $N$ as functors from that category to itself. It can be thought of as a topological group with $\pi_0$ isomorphic to $\text{Out}(N)$ and $\pi_1$ isomorphic to $Z(N)$.

The standard story in group cohomology makes extra simplifying assumptions to avoid the terror of having to talk about 2-groups. The easiest extra assumption is to assume that $N$ is central (and in particular abelian). This turns out to free us from having to think about $\text{Out}(N)$, and now the resulting central extensions are classified by the ordinary group cohomology

$$[BG, B^2 N] \cong H^2(G, N).$$

This whole story is better thought of as an aspect of homotopy theory. Short exact sequences of groups correspond to fiber sequences

$$BN \to BX \to BG$$

of homotopy types, and in general the classification of fiber sequences with fiber (homotopy equivalent to) $F$ (here $BN$) and base $B$ (here $BG$) is given by the nonabelian cohomology

$$[B, B \text{Aut}(F)]$$

of $B$ with coefficients in the topological group of self-homotopy equivalences of $F$. See also principal bundle.

Answer 2

Let $H,K$ be subgroup of $G$ such that $G=HK$ and $H\cap K=e$.

case 1: $H,K$ are normal in $G$.

In that case as we can easily show that $hk=kh$ for $h,k\in H,K$. That is why giving meaning $hk$ to $(h,k)$ is very natural.

case 2: Now it is time ask what if only $H$ is normal? Now we can not say $hk=kh$ but we can say that $H^k=H$ or we have an homomorphism $\phi: K\to Aut(H)$ and surprisingly $\phi$ completletly determines the result of "$hk$" and vice versa and $K$ is also normal if and only if $\phi$ is trivial homomorphism.

case 3: Now we removed all normality case which is known as Zappa–Szép product which define general method to define "$hk$".