Yes, your argument is correct.
To identify , look carefully at your cosets. I think that there are only two.
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.
then for some
then since where
Since , then
so since A was arbitrary!
Does this effectively show that (H is normal in G)?
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!
Are the two cosets,
matrices with positive nonzero determinant
and matrices with negative nonzero determinant?
Would that possibly be isomorphic to under addition?
I feel like this can't be the case...
Thank you for confirming the normal subgroup test. I thought I understood it in class until I tried applying it, then I was unsure! Thanks.