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زمرة (Ar); Groupe (Fr); Gruppe (Ge); Gruppo (It); (Ja); Группа (Ru).Grupo (Sp).

A set G equipped with a binary operation *: G × GG, assigning to a pair (g,h) the product g*h is called a group if:

  1. The operation is associative, i.e. (a*b)*c = a*(b*c).
  2. G contains an identity element (neutral element) e: g*e = e*g = g for all g in G.
  3. 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 point groups are finite groups.

See also

  • Chapter 1.1 of International Tables for Crystallography, Volume A, 6th edition