Subgroup

A subgroup is defined within a group $G$ with operation $\ast$. We say that $H$ is a subgroup of $G$ if it is a subset of $G$, whilst being a group itself (with the same operation as $H$). The notation we use is $H\le G$, and sometimes say that $G$ is a supergroup of $H$.

We can check this quickly using the Quick Subgroup Test. Given that $H$ is a subset of $G$, where $G$ is a group with operation $\ast$, then $(H,\ast )$ is a subgroup of $G$ if and only if:

1. $H$ contains the identity element, i.e., $e_{G\in H}$.
2. $H$ is closed under $\ast$, i.e., for all $h_1,h_2\in H$ we have $h_1\ast h_2\in H$.
3. Every element of $H$ has an inverse in $H$, i.e., for all $h\in H$ we have $h^{-1}\in H$.

Given this, we can prove the following properties:

1. The identity element in $H$, $e_H$, is the same as the identity element in $G$, $e_G$.
2. For all $h\in H$, the inverse of $h$ in $H$ is equal to the inverse of $h$ in $G$.
3. $|H|=1$ if and only if $H=⟨e_G⟩$ (the trivial group)
4. If $|G|$ is finite, then $|H|=|G|⟺H=G$.