Group

Groupe (Fr); Gruppe (Ge); Grupo (Sp); Gruppo (It); 群 (Ja).

A set G equipped with a binary operation *: G x G → G, 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

• Section 8.1.3 in the International Tables for Crystallography, Volume A