Why do we introduce groups using division? I am only starting to really study algebra so I apologize if this is an ill-formed question. When learning about groups, why is division used so heavily in the beginning? Would it not be simpler to generalize first the notion of addition or multiplication? Division is something generally added to groups to form rings, so why is it in the first paragraphs of every algebra textbook? 
I feel like I must not be seeing something glaringly obvious.
 A: I guess your question should probably be : why do we not study monoids first (monoids are like groups, but without the hypothesis that elements have an inverse) ?
And a reason is that, in addition to the fact that groups are usually considered more "important" than mere monoids, the study of monoids is actually more complicated than the study of groups. The existence of inverses is really a simplification, not a complexification. So since in the end you will most likely be interested in groups, learning first the more involved theory of monoids is probably counter-productive.
A: This might help. 
The terms "addition/multiplication/division" really don't have anything to do with the definition of groups. They are all just aliases for some familiar binary operations. Mostly they influence our notation for the operation of the group (so that it is juxtaposition, a dot, or +, or something else.)
A groups operation and its elements' inverses can be written in a multiplicative manner or additive manner. Of the two, + is usually reserved for operations that are Abelian, so it is not a good choice for an introductory notation if you intend to go on to talk about nonabelian groups. That is why you most often see groups written in multiplicative style using juxtaposition (that, and it simply takes less symbols.)
A: I am unfamiliar with anyone but a single nutjob that would go about division rather than multiplication first.
Notationally however one may use division, in a group with multiplicative notion we might write $a/b$ to mean $a\cdot b^{-1}$ as nothing but a shorthand.
Additive notion for groups is usually reserved for abelian groups.
