Difference between revisions of "Subgroup"
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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]] | [[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 1G 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.
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