# Has the external knit product been used to construct a previously unknown group?

In the Wikipedia article Zappa–Szép product , the knit product (a.k.a. Zappa–Szép product, Zappa–Rédei-Szép product, general product, exact factorization) is defined, and its basic properties are laid out. Within that article lies a section entitled "External Zappa–Szép products" which details how to take groups $$H$$ and $$K$$ (when the groups meet certain properties and when certain functions $$H\times K\rightarrow H$$ and $$H\times K\rightarrow K$$ are defined) and define a multiplication on the Cartesian product $$H\times K$$, resulting in a group which can be seen to be the internal knit product of subgroups $$H\times 1$$ and $$1\times K$$, thus constructing the external knit product. The external version, though, looks to be rather unwieldy to use in practice, and this brings me to my question.

Was a group ever discovered via the external knit product construction, that otherwise was previously unknown and where neither subgroup was normal?

It is quite likely a real pain to construct the left and right actions (i.e., the aforementioned functions $$H\times K\rightarrow *$$) and then show all caveats are met, without a priori having a group there to serve as the roadmap. Hence, if this actually has happened, say to construct a counterexample to a given conjecture, then studying how the conjecture gave rise to the actions would be (for me, at least) a fascinating study.

As an aside, there are plenty of examples wherein a group factors as an internal knit product of subgroups, neither of which is normal. In fact, given an arbitrary group $$G$$, a Frattini argument applied to the right regular action of $$G$$ on itself provides a knit product factorization of the symmetric group $${\rm Sym}(G)$$ as $$G$$ times $${\rm Sym}(G\smallsetminus 1)$$. Thus, all but the smallest such groups $$G$$ provide the specified factorizations.

• There is a related kind of situation. I am thinking of the Kegel-Wielandt Theorem, which asserts that if a finite group $G$ has a factorization $G = AB$ with $A,B$ nilpotent, then $G$ is solvable. Here, one knows nothing else a priori about $G,$ except that it has a special factorization. This could be seen as a non--existence theorem (ie, a non-solvable group can't have such a factorization). While it is not assumed that $A \cap B = 1,$ the most interesting case is when $A$ and $B$ have coprime orders. Apr 16, 2019 at 20:40
• I've never seen any such construction. Actually I have always found all this "product" terminology a bit presumptuous and "exact factorization" (or "knit factorization") better reflects what it is.
– YCor
Apr 16, 2019 at 20:52
• @GeoffRobinson It's worth pointing out it is still unknown whether the Kegel-Wielandt Theorem generalizes to infinite groups, although it has been proved in a number of special cases. Apr 16, 2019 at 22:11