$U(n)=SU(n)\rtimes U(1)$?

Question

1. Wiki says that the group $U(n)$ is a semi-direct product of $SU(n)$ and $U(1)$.

2. Each element $g$ of a semi-direct product $G=HK$, should be uniquely represented as $g=hk$.

3. $SU(n)$ and $U(1)$ have common elements, $\operatorname{e}^{i \frac{m}{n}2\pi} \mathbf{1}$. This allows for multiple expressions for elements.

What am I doing wrong?

Answer 1

According to the same wikipedia article:

Here the $\mathrm{U}(1)$ subgroup of $\mathrm{U}(n)$ can be taken to consist of matrices that are diagonal, with $\mathrm{e}^{\mathrm{i}θ}$ in the upper left corner and $1$ on the rest of the diagonal.

This copy of $\mathrm{U}(1)$ and $\mathrm{SU}(n)$ intersect only at the identity.