Burckhardtian determination of the space groups. II.
The central concept of part II is given by the congruence introduced by Frobenius in 1911 for characterizing the properties of systems of nonprimitive translations in space groups. The Frobenius congruences also play an important role in the Burckhardtian derivation of the 3-dimensional space groups. These are in fact crossed homomorphisms with an equivalence relation given by principal crossed homomorphisms. This leads to the first cohomology group of integral matrix groups (representatives of the arithmetic classes discussed in part I [see the preceding review Zbl 1062.20053]) with values in \(\mathbb{R}/\mathbb{Z}\) and to the algorithm of Zassenhaus (1948) for the derivation of space groups. For getting the affine inequivalent space groups, which also are non-isomorphic as abstract groups, one needs the normalizer of these first cohomology groups. The 3-dimensional tetragonal space groups are used as illustrative example.
scientific article; zbMATH DE number 2102273
scientific article