Difference between revisions of "Subgroup"

From Online Dictionary of Crystallography

m (See also: ITA 6th edition)
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*[[Normal subgroup]]
*[[Normal subgroup]]
*Section 8.3.3 in the ''International Tables for Crystallography, Volume A''
*Section 1.7.1 in the ''International Tables for Crystallography, Volume A'', 6<sup>th</sup> edition
[[Category:Fundamental crystallography]]
[[Category:Fundamental crystallography]]

Revision as of 16:52, 11 April 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