TITLE: Has this "generalized semidirect product" been studied?
QUESTION [20 upvotes]: If $G$ is a finite group with subgroups $H$ and $K$ such that $HK = G$ and $H\cap K = \{1\}$ we get that every element of $G$ can be written uniquely as $hk$ with $h\in H$ and $k\in K$. This then gives a map from $H\times K$ to itself sending $(h,k)$ to the unique $(h',k')$ such that $kh = h'k'$.
Conversely, given a map $\varphi$ from $H\times K$ to itself with suitable properties, one can define a group structure $G$ on $H\times K$ such that $H$ and $K$ are subgroups with $HK = G$ and $H\cap K = \{1\}$ and such that the multiplication is given via the map as $(h,k)(h',k') = (h\varphi_1(h',k),\varphi_2(h',k)k')$ where $\varphi_1$ and $\varphi_2$ are the projections on the first and second coordinate.
Has this construction been studied before? It seems like it would be a natural generalization of a semidirect product, but I have never seen it mentioned anywhere.

REPLY [2 votes]: These are also called matched pairs of groups. I just did a google search on this term and it gave over 11,400 results, often related to Hopf algebras. The n-fold case is discussed in arXiv:1201.0059 . But I was not aware of the older references given above, which are nice to see.