I came to this problem when doing the exercise that the polydisc $\Delta(0,1)^n=\prod\limits_{n}\Delta(0,1)$ in $\mathbb{C}^n$ is not biholomorphic to the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$.We can show that the group of automorphisms of $\Delta(0,1)^n$ preserving the origin is the semidirect of $U^n(1)=\prod\limits_{n}U(1)$ and the permutation group $S_n$(the semidirect product here means that $U^n(1)$ si a normal subgroup of $U^n(1)\rtimes S_n$), while that of $\mathbb{B}^n$ is $U(n)$.
Then I do not know how to show that the above two groups are not isomorphic.Intuitively, topologically the (real) dimension of $U^n(1)\rtimes S_n$ is $n$ , while that of $U(n)$ is $\frac{n(n-1)}{2}$ (is it right?), so they are not isomrphic.
Clearly it is not a rigorous proof.
Will someone be kind enough to give me some hints for this problem?Thank you very much!