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.
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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 $SO(2n)$? 

