$U(n)$ is not isomorphic to $SU(n) \times S^1$

Question

I managed to prove this after looking at proofs that $O(n)$ is not isomorphic to $SO(n) \times \pm I$ for n even by showing that the centres are of a different size.

$Z(U) = Z(SU) = λI$ but $Z(SU(n) \times S^1) = λI \times [0,2π)$. Therefore $Z(U) \subset Z(SU(n) \times S^1)$ and hence not isomorphic.

Thing is, we never discussed the centres of these groups and it seems a bit weird for a linear algebra class to have a question with a proof like that. Is there any other more linear algebra-y way to prove this?

Thanks!