The mathematical background of the subgroup tables

Abstract

This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. The algebraic concepts of vector spaces, the affine space and the affine group are defined and discussed. A section on groups with special emphasis on actions of groups on sets, the Sylow theorems and the isomorphism theorems follows. After the definition of space groups, their maximal subgroups are considered and the theorem of Hermann is derived. It is shown that a maximal subgroup of a space group has a finite index and is a space group again. From the proof that three-dimensional space groups are soluble groups, it follows that the indices of their maximal subgroups are prime powers. Special considerations are devoted to the subgroups of index 2 and 3. Furthermore, a maximal subgroup is an isomorphic subgroup if its index is larger than 4. In addition, more special quantitative results on the numbers and indices of maximal subgroups of space groups are derived. The abstract definitions and theorems are illustrated by several examples and applications. Keywords: Abelian groups; CARAT; Euclidean affine spaces; Euclidean groups; Euclidean metrics; Euclidean vector spaces; Hermann’s theorem; Lagrange’s theorem; Sylow’s theorems; affine groups; affine mappings; affine spaces; automorphism; characteristic subgroups; congruence; factor groups; faithful actions; isomorphism; generators; groups; homomorphism; klassengleiche subgroups; cosets; isomorphic groups; mappings; normalizers; space groups; isomorphism theorems; translation subgroups; maximal subgroups; translationengleiche subgroups; vector spaces; orbits; alternating groups; symmorphic space groups