Let $U(n)$ be a the unitary group the quotient $$\frac{U(n)}{U(n-1)} \simeq S^{2n-1}$$ and I believe that $$\frac{SU(n)}{SU(n-1)} \simeq S^{2n-1}.$$
Obviously the $U(n-1)$ is not a normal subgroup of $U(n)$. The $SU(n-1)$ is not a normal subgroup of $SU(n)$.
What are the maximal normal subgroups in $U(n)$ and $SU(n)$ respectively?
Write the maximal normal subgroups in $U(n)$ as $G_{U(n)}$, then what is the quotient group $U(n)/G_{U(n)}=$?
Write the maximal normal subgroups in $SU(n)$ as $G_{SU(n)}$, then what is the quotient group $SU(n)/G_{SU(n)}=$?