General Linear Groups

Very quickly, before we move on to isomorphisms like I promised, I wanted to throw in a quick plug for the general linear groups. The general linear groups are the matrix groups. I'm just going to introduce this very important group and leave out the details and the proofs because it needs some linear algebra that I'm not going to derive. The only reason I'm going over this group is because it is an excellent group that gives a lot of nice examples. Whenver I use it in the future, though, I'll make sure that if you skip it you won't miss anything since beginners may not know any linear algebra.

Definition: GL(n,F)
Let n be a positive integer and let F be any field. The set GL(n,F) is the set of all n-by-n matrices with elements in F and with a non-zero determinant.

I haven't yet defined a field, but if you don't know what a field is, just replace any field with the real numbers, one of the most common fields. If we just consider the real numbers, R, then GL(n,R) is the set of all of the standard n-by-n matrices that we've always used in linear algebra. As it turns out, this set forms a group under matrix multiplication.

Theorem: General Linear Group
GL(n,F) forms a group under the operation of standard matrix multiplication.

I'm not going to prove this theorem, but as an example, it is pretty clear that GL(n,R) is a group if you're familiar with linear algebra. Every matrix in GL(n,R) is square of dimension n so it has to be closed, all matrix multiplication is associative, the standard identity matrix is the identity element of GL(n,R), and the inverse of any matrix is the standard matrix inverse from linear algebra.

As it turns out, this is a very important and a very cool group. If you don't know enough linear algebra to get this group yet, don't worry about it. I just wanted to throw it out there so that I can use it sparringly in examples later if I want. Tune in next time for an introduction to isomorphisms.

References

Previous Related Post: Properties of Symmetric Groups

Text Reference: Gallain Chapter 2

Wikipedia: General Linear Group