Zappa–Szép product
It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H.[1] Let G be a group with identity element e, and let H and K be subgroups of G. The following statements are equivalent: If either (and hence both) of these statements hold, then G is said to be an internal Zappa–Szép product of H and K. Let G = GL(n,C), the general linear group of invertible n × n matrices over the complex numbers.For each matrix A in G, the QR decomposition asserts that there exists a unique unitary matrix Q and a unique upper triangular matrix R with positive real entries on the main diagonal such that A = QR.One of the most important examples of this is Philip Hall's 1937 theorem on the existence of Sylow systems for soluble groups.This shows that every soluble group is a Zappa–Szép product of a Hall p'-subgroup and a Sylow p-subgroup, and in fact that the group is a (multiple factor) Zappa–Szép product of a certain set of representatives of its Sylow subgroups.In 1935, George Miller showed that any non-regular transitive permutation group with a regular subgroup is a Zappa–Szép product of the regular subgroup and a point stabilizer.He gives PSL(2,11) and the alternating group of degree 5 as examples, and every alternating group of prime degree is also an example.To motivate this, let G = HK be an internal Zappa–Szép product of subgroups H and K of the group G. For each k in K and each h in H, there exist α(k, h) in H and β(k, h) in K such that kh = α(k, h) β(k, h).Turning this around, suppose H and K are groups (and let e denote each group's identity element) and suppose there exist mappings α : K × H → H and β : K × H → K satisfying the properties above.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. If H is normal in G, then the mappings α and β are given by, respectively, α(k,h) = k h k− 1 and β(k, h) = k. This is easy to see becauseIn this case, G is an internal semidirect product of H and K. If, in addition, K is normal in G, then α(k,h) = h. In this case, G is an internal direct product of H and K. Complement (group theory)