Let G be a group and H a non-empty subset of G. Then H is called a
- 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
- Section 8.3.3 in the International Tables for Crystallography, Volume A