# Difference between revisions of "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''). |

− | Let G be a group and H a non-empty subset of G. Then H is called a '''subgroup''' of | + | 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''; | ||

+ | # ''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. | + | 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. | + | 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== | ==See also== | ||

− | *[[ Normal subgroup]] | + | *[[Complex]] |

− | * | + | *[[Coset]] |

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

+ | *[[Supergroup]] | ||

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

+ | |||

+ | [[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