# Group

### 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 point groups are finite groups.

## See also

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