# Difference between revisions of "Subgroup"

### From Online Dictionary of Crystallography

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BrianMcMahon (talk | contribs) m (Tidied translations.) |
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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''). |

− | 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 | + | 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<sub>G</sub>'' of ''G'' is contained in ''H''; | # the identity element ''1<sub>G</sub>'' of ''G'' is contained in ''H''; | ||

# ''H'' is closed under the group operation (inherited from ''G''); | # ''H'' is closed under the group operation (inherited from ''G''); | ||

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*[[Normal subgroup]] | *[[Normal subgroup]] | ||

*[[Supergroup]] | *[[Supergroup]] | ||

− | * | + | *Chapter 1.7.1 of ''International Tables for Crystallography, Volume A'', 6th edition |

[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Latest revision as of 08:59, 20 November 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