Über isomorphe Untergruppen von Raumgruppen der Kristallklassen 4, , 4/m, 3, , 6, und 6/m

Number theory is used to derive which indices of symmetry reduction can occur for maximal isomorphic subgroups of space groups belonging to the crystal classes mentioned in the title and having unit cells with enlarged base vectors a and b. In the case of the crystal classes 4, and 4/m, the possible index values are i = p² with p 3 (mod 4), i = 2 and i = p I (mod 4) (p = prime number). In the crystal classes 3, , 6, and 6/m, i = p² with p 2 (mod 3), i = 3 and i = p 1 (mod 3) are possible. The number of isomorphic subgroups of index i (maximal and non-maximal) can be calculated with the formula R(i) = Σ χ_D(m), where m runs through all divisors of i and χ_D(m) is the Dirichlet character mod |D|; D = −4 for the tetragonal and D = −3 for the trigonal and hexagonal space groups. χ₋₄(m) is equal to 0 for m 0 (mod 2), 1 for m = p 1 (mod 4), −1 for m = p 3 (mod 4), and the corresponding product for nonprime values of m. χ₋₃(m) is equal to 0 for m 0 (mod 3), 1 for m = p 1 (mod 3), −1 for m = p 2 (mod 3), and their corresponding product for nonprime m. R(i) is the number of conjugacy classes, each of which comprises i conjugate subgroups (for i > 2).