From Online Dictionary of Crystallography
زمرة (Ar). Groupe (Fr). Gruppe (Ge). Gruppo (It). 群 (Ja). Группа (Ru). Grupo (Sp).
A set G equipped with a binary operation *: G × G → G, assigning to a pair (g,h) the product g*h is called a group if:
- The operation is associative, i.e. (a*b)*c = a*(b*c).
- G contains an identity element (neutral element) e: g*e = e*g = g for all g in G.
- Every g in G has an inverse element h for which g*h = h*g = e. The inverse element of g is written as g−1.
Often, the symbol for the binary operation is omitted. The product of the elements g and h is then denoted by the concatenation gh.
The binary operation need not be commutative, i.e. in general one will have g*h ≠ h*g. In the case that g*h = h*g holds for all g,h in G, the group is an Abelian group.
A group G may have a finite or infinite number of elements. In the first case, the number of elements of G is the order of G. In the latter case, G is called an infinite group. Examples of infinite groups are space groups and their translation subgroups, whereas crystallographic point groups are finite groups.
- Chapter 1.1 of International Tables for Crystallography, Volume A, 6th edition