A subgroup of a Group $(G,*)$ 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_element inverse]. We get associativity for free because $G$ is a group. So the requirements of a subgroup $H$ are:

1. Closure: For any $x, y$ in $H$, $x*y$ is in $H$.
2. Identity: The identity $e$ of $G$ is in $H$.
3. [inverse_element 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]).