Why is abelianness such a precious property? My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic happens" when speaking of this property, or qualities that mimic commutativity in some way. So why is it such a game-changing quality?
Thanks!
 A: For one, in studying an abelian group you can utilize a lot of your intuition from doing addition.  
But more to the point about normal subgroups - One should note that in an abelian group $G$, every subgroup $H$ is normal, so you can always take the quotient $G/H$.  In a non-abelian group, you need special conditions on $H$.  The definition of a normal subgroup is a subgroup  $H$ such that when $h\in H$, $ghg^{-1}\in H$ for every $g\in G$.  You can see how this condition is trivial when G is abelian.  But in the non-abelian case it's just what is needed for the quotient $G/H$ to be a group.  I think this might be what your teacher meant when he said normal subgroups are "trying to capture a little bit of abelianness" - they do something that every subgroup can do in an abelian group.    
Of course, this really doesn't even start the whole picture.  I can't talk about every example, but here, in a nutshell, is one I've been trying to understand: Let $K$ be a number field.  It turns out that the field $K$ itself (in particular, it's group of fractional ideals) has enough information to describe all of it's abelian extensions, that is, the extensions whose galois group is abelian.  What about the non-abelian extensions?  Oh, we don't even know.  That's an open research area.
Many times, groups can be attached to another more complicated object (like a number field, in the above example) to tell us something about that object.  When these groups are abelian, it often means these objects are particularly nice and well-behaved.  
But we also just know a lot more about abelian groups themselves.  For instance, every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order.  What can we say about every finite non-abelian group?  We don't know them all, but we know they can get really complicated
