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”.
Let’s consider a homomorphism .
The image of — written — is the collection of all elements of that actually occur as values of . If and are such elements, then , so it’s also in the image of . Check for yourself that the image is closed under inverses and that it contains the identity of . This shows that is a subgroup of .
The kernel of — written — is the collection of all elements of that are sent to the identity of . If and are in the kernel, then , so is also in the kernel of . Again, verify for yourself that the kernel is also closed under inverses and contains the identity of . This shows that is a subgroup of .
Go back to last week’s post about homomorphisms and figure out the image and kernel of each of the examples. Also consider the following questions
- What is the kernel of a monomorphism?
- What is the image of an epimorphism?