Abstract
As a foundation for Klein's fundamental idea about the connection of geometry and its motion group and the unified description of all Cayley-Klein geometries, a method of group transitions including contractions as well as analytical continuations of the groups is developed. The generators and Casimir operators of an arbitrary Cayley-Klein group are obtained from those of the classical orthogonal group. The classification of all possible transitions between the Cayley-Klein groups is given. The physically important case of the kinematic groups is discussed.
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