# Subgroup

### From Online Dictionary of Crystallography

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- | <font color="blue">Sous-groupe</font> (''Fr'') | + | <font color="orange">زمرة جزئية</font> (''Ar''); <font color="blue">Sous-groupe</font> (''Fr''); <font color="red">Untergruppe</font> (''Ge''); <font color="black">Sottogruppo</font> (''It''); <font color="purple">部分群</font> (''Ja''); <font color="brown">Подгруппа</font> (''Ru''); <font color="green">Subgrupo</font> (''Sp''). |

## Revision as of 15:29, 10 October 2017

زمرة جزئية (*Ar*); Sous-groupe (*Fr*); Untergruppe (*Ge*); Sottogruppo (*It*); 部分群 (*Ja*); Подгруппа (*Ru*); Subgrupo (*Sp*).

Let *G* be a group and *H* a non-empty subset of *G*. Then *H* is called a **subgroup** of *G* if the elements of *H* obey the group postulates, *i.e.* if

- the identity element
*1*of_{G}*G*is contained in*H*; -
*H*is closed under the group operation (inherited from*G*); -
*H*is closed under taking inverses.

The subgroup *H* is called a **proper subgroup** of *G* if there are elements of *G* not contained in *H*.

A subgroup *H* of *G* is called a **maximal subgroup** of *G* if there is no proper subgroup *M* of *G* such that *H* is a proper subgroup of *M*.

## See also

- Complex
- Coset
- Normal subgroup
- Supergroup
- Chapter 1.7.1 of
*International Tables for Crystallography, Volume A*, 6th edition