Crystal symmetry groups and their representations I lattice extensions of point groups in R 2 and R 3

Abstract

We classify all point groups in R 2 and R 3 according to their decomposition into m-fold semidirect products of cyclic groups. This product is characterized completely by means of m−1 matrices α (1),…, α ( m−1) with non-negative integer elements. For any arithmetic class [ P n i ] in R n ( n=2, 3) we define the set D( P n i , T) of representative translations with the following properties: (a) The elements of D( P n i , T) are in one-to-one correspondence with defining relations of P n i ; (b) D( P n i , T) determines uniquely all equivalence classes of the factors of P n i over T. We present: (i) a construction of all sets D( P n i , T) in R 2 and R 3; (ii) a construction of all algebraic classes of space groups and a derivation of all group operations which transform one algebraic class into another; (iii) a derivation of formulas expressing the multiplication table of any space group by means of the representative translations and matrices α (1),…, α ( m−1) . The results (i), (ii) and (iii) are obtained without making use of the notion of non-primitive translations.