Prove or disprove that $H$ is a subgroup of $C^*$ under multiplication.
Let $$H = \{a + bi \,|\, a, b R, a^2 + b^2=1\}$$ I know for every $a \in H$, if we can prove its inverse is in $H$ then we are done.
I said let $a+bi$ and $c+di$ belong to $H$. Then $$(a + bi)(c + di)^{-1} = (ac + bd) + (bc - ad)i$$ and $$(ac + bd)^2 + (bc - ad)^2 = 1$$
Hence $H$ is a subgroup of $C^*$ under multiplication.
Yes? No? How can I be more detailed in my answer? I do have to teach this back to someone. Thank you.