A subgroup of a group $(G,\ast )$ is a group of the form $(H,\ast )$, 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:

1. Closure: For any $x,y$ in $H$, $x\ast y$ is in $H$.
2. Identity: The identity $e$ of $G$ is in $H$.
3. 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).