# 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.

• What part don't you understand - do you need help with the definitions of $\mathrm{U}(n)$ and $\mathrm{SU}(n)$? – Zev Chonoles Jul 22 '13 at 4:12
• Yes, there is no question here. – Thomas Andrews Jul 22 '13 at 4:13
• @ZevChonoles no I have those definitions, could you help me show that its isomorphic to U(1) – user57875 Jul 22 '13 at 4:17

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.