Abstract
Taking into account the fact that space groups are groups of transformations of Euclideann-dimensional space, non-equivalent systems of non-primitive translations are defined. They can be brought into one-to-one correspondence with the elements of the groupH 1 (K, R n/Zn) or with those of the groupH 1 (K, Z n/kZn)/H 1 (K, Z n). (K is a point group of orderk.) The consistency of these findings with the results of Part I is given by the isomorphisms
Theorems are proved giving the conditions for cohomology groupsH q (K, A) to be zero. These conditions are fulfilled in particular ifA=R n andK is a subgroup ofGL (n, R) that either is compact (thenq>0) or has a finite normal subgroup leaving no element ofR n invariant (thenq≧0). This implies that the affine, the Euclidean and the inhomogeneous Lorentz groups are the only extensions ofR n by the corresponding homogeneous groups. By way of illustration, the theory of this paper is applied to two 2-dimensional space groups.
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Ascher, E., Janner, A. Algebraic aspects of crystallography. Commun.Math. Phys. 11, 138–167 (1968). https://doi.org/10.1007/BF01645902
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DOI: https://doi.org/10.1007/BF01645902