Example of Group: GL(2,R) (2 of 3)

Abstract Algebra: Let G = GL(2, R) be the group of real 2 x2 invertible matrices, and let H be the subset of matrices with determinant = 1. We show that H is a normal subgroup of G directly and by exhibiting H as the kernel of a homomorphism.

[MathDoctorBob]

You're welcome! If you are looking for more group theory, I have a semester long playlist with problem sets and solutions (at the website). - Bob

[MathDoctorBob]

One way is by trying to find all matrices that commute with all diagonals of 1s and -1s, and see which are in O(n). Start with 2x2 and 3x3 cases.

If you know canonical forms, we can put every matrix in O(n) in special form by conjugation by another element of O(n). If in the center, it must already be in this form since conjugation does nothing. Special form is diag (1, -1, r(theta)), where we have a list of 1s, a list of -1s, and 2x2 blocks which represent rotations by thetas. - Bob

[MathDoctorBob]

You're welcome! What did you use to find the center of O(n)? - Bob

[MathDoctorBob]

Ah, different questions. That proof fails when n is even because -I is in SO(n). (det(-I) = (-1)^n).

If we could find an isomorphism to SO(n) x Z2, then we would have an element in the center of O(n) corresponding to (0, 1), but not in SO(n). But the center is I, -I and contained in SO(n).

In fact, for n even, if J=diag(-1, 1, 1, ..) then every element of O(n) is of the form A or JA where A is in SO(n).

For the first part, simultaneous diagonals? Canonical form for O(n)? - Bob

[MathDoctorBob]

We need closure under multiplication. So if A and B have det =2, then det(AB) = det(A)det(B) = 4, and AB is not in the set with det = 2. Hope that helps!

[MathDoctorBob]

I haven't used it, but it sounds like a hard book for beginners. The one I usually send students to is Fraleigh's A First Course in Abstract Algebra. It's not necessarily the best book for this material, but he does a great job for students new to the material.

-Bob

[MathDoctorBob]

You're welcome! Glad to be of help.

[MathDoctorBob]

What kind of results are you allowed to use? That's going to affect the answer, as there are a few ways to approach the problem. The answer to the first part is {I, -I}. When n is odd, O(n) = SO(n) x Z/2 as every element is in the form A or (-I)A and {I, -I} n SO(n) = I. This will fail for n even since -I is in SO(n), but we need an argument. More later. -Bob

[alancristopher]

How Prove that GL(2, R)/SL(2, R) is isomorphic to R* ?

[dominiqueperron]

i was working on the cyclic group and the generator of SL(2, R), however im not to confident i did it correctly. Could I get any help?

[janetkonyu]

how do you show a matrix group is abelian