Normal Subgroup of the General Linear Group?

Hello,

I am trying to show that the group is a normal subgroup of , where

I am just wondering if this proof is in the correct style (Normal Subgroup Test).

So, let be an arbitrary matrix in

then, is also in

now, we want to show to show that (H is normal in G). This is the normal subgroup test.

so let

then for some

then since where

Since , then

So

so since A was arbitrary!

Does this effectively show that (H is normal in G)?

Thank you!!

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Also, if the above is correct,

I am having trouble with the second part of the question, which asks:

Which common, known group is isomorphic to the factor group G/H for the groups G,H in the above question?

(the factor group G/H is defined as the group of left-cosets

I can't see right away that the factor group G/H is isomorphic to anything that I could think of!

Thanks!!