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'').  

Revision as of 15:29, 10 October 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