Zappa–Szép product

In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei-Szép product, general product, knit product or exact factorization) 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 (1940) and Jenő Szép (1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).

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. Thus G is a Zappa–Szép product of the unitary group U(n) and the group (say) K of upper triangular matrices with positive diagonal entries.

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