A subgroup is pretty straightforward. It’s a little group living inside a bigger group. If you’ve got a group and some collection of elements of so that is a group using the same composition as , then is a subgroup. To be more explicit, you need that
- If and are in then is in .
- If is in then is in .
- The identity of is in . [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 and take compositions and inverses we never leave the subgroup.
The collection of all even integers is a subgroup of the group of all integers (with addition as the operation).
The subset is a subgroup of the group .
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”.