Let be a group and any subset of . Let be any two elements of . Now being a member of the product of surely belongs to , but it may or may not belong to . If, however, belongs to , we say that is stable for the composition in and the composition in has induced the composition in . If is itself a group for the induced composition, then we say that is a subgroup of .
Definition: A non-empty subset of a group is said to be a subgroup of if the composition in induces a composition in and if is a group for the induced composition.
The two subgroups (i) consisting of the identity element alone, and (ii) the group itself are always present in a group .
There is, however, a trivial subgroup. A subgroup other than these two is known as a proper subgroup.
A complex is any subset of a group, whether it is a subgroup or not. It is easy to prove that:
(i) The identity of a subgroup is the same as that of a group.
(ii) The inverse of any element of a subgroup is the same as the inverse element regarded as a member of the group.
(iii) The order of any element of a subgroup is the same as that of the element regarded as a member of the group.
(i) The additive group of integers is a subgroup of the additive group of rational numbers.
(ii) The multiplicative group of positive rational numbers is a subgroup of the multiplicative group of non-zero real numbers.
(iii) The multiplicative group is a subgroup of the multiplicative group .