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Difference between revisions of "Subgroup"

From Online Dictionary of Crystallography

m (See also: ITA 6th edition)
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<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'').
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<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
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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]]
*Section 1.7.1 in the ''International Tables for Crystallography, Volume A'', 6<sup>th</sup> edition
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*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

  1. the identity element 1G of G is contained in H;
  2. H is closed under the group operation (inherited from G);
  3. 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