From Online Dictionary of Crystallography
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
- the identity element 1G of G is contained in H;
- H is closed under the group operation (inherited from G);
- 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.