# Subgroups

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.

__Examples__**:**

**(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 .