# The Unapologetic Mathematician

## Subgroups

A subgroup is pretty straightforward. It’s a little group living inside a bigger group. If you’ve got a group $G$ and some collection $H$ of elements of $G$ so that $H$ is a group using the same composition as $G$, then $H$ is a subgroup. To be more explicit, you need that

• If $x$ and $y$ are in $H$ then $xy$ is in $H$.
• If $x$ is in $H$ then $x^{-1}$ is in $H$.
• The identity $e$ of $G$ is in $H$. [added at the suggestion of Toby Bartels]

We say that a subgroup is “closed” under composition and inverse, meaning that if we start with elements of $H$ and take compositions and inverses we never leave the subgroup.

The collection of all even integers is a subgroup of the group ${\mathbb Z}$ of all integers (with addition as the operation).
The subset $\{e,(1\,2\,3),(1\,3\,2)\}$ is a subgroup of the group $S_3$.
Every group has two “trivial” subgroups: the whole group itself, and the subgroup consisting of just the identity element.
There are two ways of getting subgroups that I want to spend a bit more time on: “images” and “kernels”.