High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, especially group theory, the Zappa–Szep product (also known as the knit product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa and Jeno Szep. Let G = GL(n,C), the general linear group of invertible n x 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. Thus G is a Zappa–Szep product of the unitary group U(n) and the group (say) K of upper triangular matrices with positive diagonal entries. One of the most important examples of this is Hall's 1937 theorem on the existence of Sylow systems for soluble groups. This shows that every soluble group is a Zappa–Szep product of a Hall p'-subgroup and a Sylow p-subgroup, and in fact that the group is a (multiple factor) Zappa–Szep product of a certain set of representatives of its Sylow subgroups.

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