Thread: Identifying Quotient Groups

Hello,

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!!

Some quick progrss....

If we define a homomorphism for the reals under multiplication , by



(I have not checked that this is actually operation preserving but I think it is...)

Then the identity of is 1, so the

which is just the group H.

Since maps onto
Then by the First Isomorphism Theorem, do we have ??

Not sure if this is solid... A second opinion would be helpful. Thanks!

Originally Posted by matt.qmar

Some quick progrss....

If we define a homomorphism for the reals under multiplication , by



(I have not checked that this is actually operation preserving but I think it is...)

Then the identity of is 1, so the

which is just the group H.

Since maps onto
Then by the First Isomorphism Theorem, do we have ??

Not sure if this is solid... A second opinion would be helpful. Thanks!

Looks good to me, as long as you do prove that that is a homomorphism.

You have a normal subgroup, N, and it is the kernel of a homomorphism. Thus, apply the first isomorphism theorem to get that,

.