Given the set G of invertible, upper triangular 2x2 matrices, it can be verified that G forms a subgroup of the general linear group (invertible matrices) of 2x2 matrices with real entries.
Now, consider the set H of matrices of the form
where are real and
Now, it can be verified that H is a subgroup of G
In fact, it can be verified (I have done the dirty work!) that H is normal in G.
So, now we are asked to identify the quotient group
That is to say, we are asked to use the First Isomorphism Theorem to show it is isomorphic to a known group.
So by my understanding, the trick is to identify some surjective homomorphism whose kernel is the group H, , and then the image of that homomorphism, , is isomorphic to the quotient group ?
I'm not sure where to start with this problem!
Any help appreciated, thanks!!