CRYSTALLOGRAPHIC GROUPS IN SPACE AND TIME .2. CENTRAL EXTENSIONS

As space groups may be obtained from extensions of a free abelian group by a finite group, the extension conditions, given for abelian groups by M. Hall, are worked out in more detail for central extensions by groups isomorphic to Euclidean point groups. For four-dimensional point groups which are (3 + 1)-reducible over R these conditions are explicitly given. For extensions of Zn a knowledge of these conditions is not necessary. For these extensions a simple method to determine all non-equivalent extensions is given. The analogy of this method to a method by Zassenhaus is explained by the isomorphism of H2φ(K, Zn) and H2φ(K, Rn/Zn). Also the determination of isomorphism classes of extensions is discussed. © 1969.