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 $\left\{ {1, - 1,i, - i} \right\}$.