I was working on the following problem from Stillwell's Naive Lie Theory.
Prove that $U(n)/Z(U(n))=SU(n)/Z(SU(n))$.
It was shown earlier in the text that $Z(U(n))=\{ e^{i\theta} \textbf{1}: \theta \in \mathbb{R} \} \cong S^{1}$ and $Z(SU(n))=\{ \omega \textbf{1}: \omega^{n}=1 \text{ and } \omega \in \mathbb{C} \}$. We have that $\textbf{1}$ denotes the identity matrix.
When thinking about this problem, I first considered the case where $n=2$. We have that unitary matrices in $U(2)$ are of the form \begin{equation*} e^{i\theta} \begin{bmatrix} \alpha &-\beta \\ \bar{\beta} & \bar{\alpha} \end{bmatrix} \end{equation*} We have that the relation which sends $e^{i\theta} \begin{bmatrix} \alpha &-\beta \\ \bar{\beta} & \bar{\alpha} \end{bmatrix}$ to $ \begin{bmatrix} \alpha &-\beta \\ \bar{\beta} & \bar{\alpha} \end{bmatrix}$ seems to be a well defined function from $U(2) \rightarrow SU(2)/Z(SU(2)$ since the center $Z(SU(2))$ consists of the matrices $\pm \textbf{1}$. This function even is a homorphism that has $Z(U(2))$ as it's kernel. From here we can conclude that $U(2)/Z(U(2))=SU(2)/Z(SU(2))$
$\textbf{However, I am having trouble generalizing from here}$