What is the quotient group of $O_2$/$SO_2$? I am working on this as a challenge problem, but I have never worked with a quotient group outside of modular values in the integers.
$SO_2$ is, by definition,  the kernel of the homomorphism $O_2\to \Bbb F^\times$,  $A\mapsto \det A$, which happens to be onto.
$SO_2$ is, by definition,  the kernel of the homomorphism $O_2\to \Bbb F^\times$,  $A\mapsto \det A$, which happens to be onto.