U(n) is the group of n by n unitary complex matrices and SO(2n) is the group of 2n by 2n real orthogonal matrices with determinant 1.So far I can show that how to get an injective group homomorphism from U(n) to SO(2n). This shows that U(n) is isomorphic to a subgroup of SO(2n).But I have no idea whether U(n) is normal in SO(2n)? If someone could explain this to me or just point out some reference for me,I will appreciate your help. Thanks.
Thanks. The answers clarify many things for me. The reason that I ask this question is that I am confused about what's the meaning of the quotient $SO(2n)/U(n)$ if $U(n)$ is not isomorphic to a normal subgroup of $SO(2n)$?