If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very lossy operation. The map G → A is always a surjection/quotient, because we can construct A by dividing G by the minimal normal subgroup that contains all conjugations ghg-1h-1 for g,h∈G.
If V is a finite-dimensional (super)vector space over a field K, then the abelianization of GL(V) is isomorphic to the multiplicative group K* of non-zero numbers in K. Indeed, the determinant exhibits the desired isomorphism.
Here are two questions I'm curious about:
- What can be said about the abelianizations of other (finite-dimensional) Lie groups?
- If V is an infinite-dimensional vector space, what can be said about the abelianization of GL(V)? Most infinite-dimensional vector spaces have some analytic structure, e.g. topological vector spaces, and so it's reasonable to ask that the operators in GL(V) should preserve that structure; you are welcome to take your favorite type of infinite-dimensional vector space and your favorite type of GL(V), if you want.