Subgroup

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 1_G 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

• Complex
• Coset
• Normal subgroup
• Supergroup
• Chapter 1.7.1 of International Tables for Crystallography, Volume A, 6th edition