ISOMORPHIC SUBGROUPS OF SPACE-GROUPS OF T HE CRYSTAL CLASSES 4, (4)OVER-BAR, 4 M, 3, (3)OVER-BAR, 6, (6)OVER-BAR AND 6/M/

Number theory is used to derive which indices of symmetry reduction ca n 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,($ ) over bar 4 and 4/m, the possible index values are i = p(2) with p = 3 (mod 4), i = 2 and i = p = 1 (mod 4) (p = prime number). In the crys tal classes 3, ($) over bar 3, 6, ($) over bar 6 and 6/m, i = p(2) wit h p = 2(mod 3), i = 3 and i = p = 1 (mod 3) are possible. The number o f isomorphic subgroups of index i (maximal and non-maximal) can be cal culated with the formula R(i) = Sigma chi(D)(m), where m runs through all divisors of i and chi(D)(m) is the Dirichlet character mod \D\; D = -4 for the tetragonal and D = -3 for the trigonal and hexagonal spac e groups. chi(-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 no nprime values of m. chi(-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 pr oduct for nonprime m. R(i) is the number of conjugacy classes, each of which comprises i conjugate subgroups (for i > 2).