Les sous-groupes isomorphes d'un groupe d'espace de type p4. I. Détermination univoque

From a given standard setting (O, A, B) (conventional unit-cell origin and vectors) of a two-dimensional space group G(p4), it is possible, for each isomorphic subgroup g(p4), to select exactly one standard setting (o, a, b) subject to the following conditions. (1) Vector conditions: a = p₁A + p₂B, b = -p₂A + p₁B, P₁ > 0, P₂ ≥ 0 (P₁, P₂ ∈ ). (2) Origin conditions: (a) if (P₁ + P₂) is odd, then the coordinates X, Y of o, with respect to (O, A, B), obey the next conditions: X, Y integers, 0 ≤ X < GCD(P₁, P₂), 0 ≤ Y < (P²₁ + P²₂)/ GCD(P₁, P₂), GCD = greatest common divisor; (b) if (P₁ + P₂)/GCD(P₁,P₂) is even, then 2X and 2Y are both even or odd, 0 ≤ X < GCD(P₁, P₂), 0 ≤ Y < (P²₁ + P²₂)/2GCD(P₁, P₂); (c) if P₁, P₂ are even and (P₁ + P₂)/GCD(P₁, P₂) is odd then 2X and 2Y are both even or odd, 0 ≤ X < GCD (P₁, P₂)/2 and 0 ≤ Y < (P²₁ + P²₂) /GCD(P₁, P₂). In any case there are exactly (P²₁ + P²₂) subgroups relevant to the same vector conditions. Tables of isomorphic subgroups p(4) are given for indices up to 25.