Difference between revisions of "Subgroup"
From Online Dictionary of Crystallography
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| − | <font color="blue">Sous-groupe</font> (''Fr'') | + | <font color="blue">Sous-groupe</font> (''Fr''). <font color="red">Untergruppe</font> (''Ge''). <font color="green">Subgrupo</font> (''Sp''). <font color="black">Sottogruppo</font> (''It''). <font color="purple">部分群</font> (''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 | + | 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]] | ||
Revision as of 11:29, 17 May 2017
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 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