# Difference between revisions of "Group"

### From Online Dictionary of Crystallography

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− | <font color="orange">زمرة</font> (''Ar'') | + | <font color="orange">زمرة</font> (''Ar''). <font color="blue">Groupe</font> (''Fr''). <font color="red">Gruppe</font> (''Ge''). <font color="black">Gruppo</font> (''It''). <font color="purple">群</font> (''Ja''). <font color="brown">Группа</font> (''Ru''). <font color="green">Grupo</font> (''Sp''). |

## Revision as of 14:43, 13 November 2017

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