# Difference between revisions of "Subgroup"

### From Online Dictionary of Crystallography

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− | 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== |

## Revision as of 10:46, 2 April 2009

Sous-groupe (*Fr*); Untergruppe (*Ge*); Subgrupo (*Sp*); Sottogruppo (*It*); 部分群 (*Ja*).

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
- Section 8.3.3 in the
*International Tables for Crystallography, Volume A*