Here $U_1$,$U_2$ and $U_3$ are unitary groups. I am interested in the generalization of this question to :
Is $U_k$ $\oplus$ $U_{(n-k)}$ a normal subgroup of $U_n$?
where $0< k < n$.
The motivation is to look at the quotient group, $$G = U_n/(U_k\oplus U_{(n-k)})$$ once it is established that the above statement holds true.
Any leads will be appreciated. Thank you!