TITLE: Normal Subgroups of $SU(n)$
QUESTION [7 upvotes]: I was wondering if there is any classification for normal subgroups of $SU(n)$?  In particular, I think that the answer is no for $n = 2$ by looking at the covering map onto $SO(3)$, but I was curious if there were any results for arbitrary $n$?
I'm actually looking for finite normal subgroups of the unitary group (a lot to ask for, I know).

REPLY [5 votes]: Qiaochi's answer deals with closed normal subgroups. However, a priori, there could have been non-closed proper normal subgroups in $SU(n)$ (not contained in its center). One can show that this cannot happen "with bare hands" (this is done for instance in Berger's "Geometry" book in the case of $SO(3)$ merely using elementary geometry), or use the following general theorem:
Theorem. Suppose that $G$ is a compact Hausdorff topological group, which is topologically simple, i.e. contains no proper closed normal subgroups. Then $G$ is simple as an abstract group. 
See Theorem 9.90 of
Karl H. Hofmann, Sidney A. Morris, The Structure of Compact Groups. A Primer for the Student – A Handbook for the Expert, de Gruyter Studies in Mathematics, Volume 25, 1998. 
Addendum. There exist Hausdorff topologically simple groups which are not simple as abstract groups, see section 3 in
George A. Willis, Compact open subgroups in simple totally disconnected groups, Journal of Algebra, Volume 312, Issue 1 (2007) pages 405-417.