A subgroup of a group is a group of the form $(H, *)$ , where $H \subset G$ . We usually say simply that $H$ is a subgroup of $G$ .

For a subset of a group $G$ to be a subgroup, it needs to satisfy all of the group axioms itself: closure, associativity, identity, and inverse. We get associativity for free because $G$ is a group. So the requirements of a subgroup $H$ are:

Closure: For any $x, y$ in $H$ , $x * y$ is in $H$ .

Identity: The identity $e$ of $G$ is in $H$ .

Inverses: For any $x$ in $H$ , $x^{- 1}$ is also in $H$ .

A subgroup is called normal if it is closed under conjugation.

The subgroup relation is transitive: if $H$ is a subgroup of $G$ , and $I$ is a subgroup of $H$ , then $I$ is a subgroup of $G$ .

Examples

Any group is a subgroup of itself. The trivial_group is a subgroup of every group.

For any integer $n$ , the set of multiples of $n$ is a subgroup of the integers (under addition).