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Difference between revisions of "Subgroup"

From Online Dictionary of Crystallography

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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="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 H if the elements of H obey the group postulates.
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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
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# the identity element ''1<sub>G</sub>'' of ''G'' is contained in ''H'';
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# ''H'' is closed under the group operation (inherited from ''G'');
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# ''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.
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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.
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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==
* [[Coset]]
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*[[Complex]]
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*[[Coset]]
 
*[[Normal subgroup]]
 
*[[Normal subgroup]]
*Section 8.3.3 in the ''International Tables for Crystallography, Volume A''
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*[[Supergroup]]
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*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

  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