Quotient of unitary group by special unitary group Can someone help me, I don't understand the following question.
"Using without proof, the homomorphism theorem, or otherwise, show that
$U(n)/SU(n)$ is isomorphic to  $U(1)$."
Here, $U(n)$ is the unitary group while $SU(n)$ is the special unitary group.

Consider the determinant map $\det:\mathrm{U}(n)\to \mathrm{U}(1)$, the kernel of which is (by definition) $\mathrm{SU}(n).$ Apply the first homomorphism theorem.
