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]).