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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.

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  • $\begingroup$ 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. $\endgroup$ Apr 16, 2019 at 20:40
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    $\begingroup$ 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. $\endgroup$
    – YCor
    Apr 16, 2019 at 20:52
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    $\begingroup$ @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. $\endgroup$
    – Derek Holt
    Apr 16, 2019 at 22:11

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