Here's sort of an "opposite" question: Is it possible to naturally and intuitively go from the idea of "groups as group presentations/quotients of the free group" to the idea of groups as symmetry groups? Certainly the reason for why abelian groups are counting things arises immediately in the first context... but of course this isn't a particularly nice place to work, since all sorts of things are undecidable.

Oh, right, it's totally possible to go from group presentations to symmetry groups, although not particularly nicely. There is, of course, a standard construction given a group presentation of a guy on which the group acts... the problem is, we either have to work with a silly definition of automorphism which nobody really uses in real life, or introduce some rather gross and unintuitive combinatorial gadgets that complicate things, or accept that there are a bunch of extra automorphisms that the presentation doesn't account for, and which in particular will generally (always?) make the automorphism group non-abelian... Is there a good way to get around this problem?

ETA: Now that I think about it, though, this isn't even necessarily the core issue with the "co-question." It's pretty easy to think about the free abelian group: It's just how we count apples and oranges if we don't want 3 apples and 3 oranges to make 6 fruits. (If you don't want to deal with negative oranges, fine; it's how you, a stock speculator, keep track of your investments [counting short sells] if you -- for good reason -- don't want to mix up your shares of Reynolds Firearms and Roslin Pharmaceuticals.) From there, it's an easy abstraction to think, well, if I lay out my apples and oranges on the table, then the order they're in doesn't matter. And then you (now you're a mathematician and not a grocer) might think, oh, but what if order does matter, which gets you to the free group. But the conceptual leap from there to general finitely presented groups is non-obvious to me. Maybe I should go back to stacking these cantaloupes...