Today I was thinking about composition of functions. It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.
But then I thought about commutativity. My first intuition was that bijective self maps of a space should commute but then I saw some counter-examples. The symmetric group is only abelian if n<=2 so clearly there needs to be more restrictions on functions than bijectivity for them to commute.
The only examples I could think of were boring things like multiplying by a constant or maximal tori of groups like O(n) (maybe less boring).
My question: In a euclidean space, what are (edit) some nice characterizations of sets of functions that commute? What about in a more general space?
Bonus: Is this notion of commutativity important anywhere in analysis?