Is SU(2) a normal subgroup of SL(2,C)? I am defining a map from SL(2/C)/SU(2) -> SL(2,C) and obviously it would be convenient if SU(2) is a normal subgroup so that I can define a group homomorphism.  However, I'm not sure whether or not it is normal.  
 A: No. More generally, $\mathfrak{sl}(2,\mathbb C)$ is simple as a real Lie algebra, and so any normal subgroup of $\mathrm{SL}(2,\mathbb C)$ must be discrete.  Indeed, the notion of "simple Lie algebra" doesn't care how big a field your Lie algebra is defined over.  This is an important example to keep in mind.  You probably know that the compact (real) simple Lie groups are in bijection with the simple complex Lie groups, but that there are other real simple Lie groups, which exist as "real forms" (fixed points under an involution) on a complex guy.  But the important thing to remember is that even if you only care about simples (and not, say, semisimples), then to classify the simples over $\mathbb R$ you must look at non-simple semisimples over $\mathbb C$.  The example here is that the complexification $\mathfrak{sl}(2,\mathbb C) \otimes_{\mathbb R} \mathbb C \cong \mathfrak{sl}(2,\mathbb C) \oplus \mathfrak{sl}(2,\mathbb C)$, which is not simple even though it has a simple real form.
In the future, I would suggest that this particular question is better suited for math.stackexchange.  Nearby questions may be appropriate for this site, but an important part in making them so is for you to include a little bit more motivation and background.  Namely, please include a little indication of what you already know (what is your level of expertise in related topics?), and also please include some discussion of how this question relates to your mathematical research.  Lie Theory certainly is a part of active professional mathematical research, but it is also a part of some undergraduate curriculums, and you will get better answers if the answerers know their audience.
