The holomorphic automorphism group of the complex disk (Q1323549)

Each holomorphic permutation of the open disk in \(\mathbb{C}\) with positive radius \(c\) and centre 0 can be written as the product of a rotation and a so-called gyration \(z\mapsto(a+z)/(1+\overline{a}z/c^ 2)\). Properties of this product decomposition are axiomatized, giving rise to the notions of a gyrogroup and a gyrosemidirect product.