# Subgroups

Let $$G$$ be a group and $$H$$ any subset of $$G$$. Let $$a,b$$ be any two elements of $$H$$. Now $$a,b$$ being a member of $$G$$ the product of $$a\,b$$ surely belongs to $$G$$, but it may or may not belong to $$H$$. If, however, $$a\,b$$ belongs to $$H$$, we say that $$H$$ is stable for the composition in $$G$$ and the composition in $$G$$ has induced the composition in $$H$$. If $$H$$ is itself a group for the induced composition, then we say that $$H$$ is a subgroup of $$G$$.

__Definition__**:** A non-empty subset $$H$$ of a group $$G$$ is said to be a subgroup of $$G$$ if the composition in $$G$$ induces a composition in $$H$$ and if $$H$$ is a group for the induced composition.

The two subgroups **(i)** consisting of the identity element alone, and **(ii)** the group $$G$$ itself are always present in a group $$G$$.

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 $$\left\{ {1, – 1} \right\}$$ is a subgroup of the multiplicative group $$\left\{ {1, – 1,i, – i} \right\}$$.