sentences:  On the cartesian product "H" × "K", define a multiplication and an inversion mapping by, respectively,
  
  
  Then "H" × "K" is a group called the external Zappa–Szép product of the groups "H" and "K".The subsets "H" × {"e"} and {"e"} × "K" are subgroups isomorphic to "H" and "K", respectively, and "H" × "K" is, in fact, an internal Zappa–Szép product of "H" × {"e"} and {"e"} × "K".Let "G" = "HK" be an internal Zappa–Szép product of subgroups "H" and "K".