I think it would be very difficult to prove things about groups using that definition. For instance, it's not obvious to me how one would even define a subgroup in this context.
I'd guess it would be a subset of the set of isomorphisms that fixes a certain subset of the graph. But maybe I'm wrong.
I don't disagree that working straight from the graph-theoretic definition might make things harder. My complaint is that maths is taught as formal definition -> theorems. What I would like to see is intuitive definition -> formal abstraction of intuition -> theorms.
In my maths degree I spent far too long asking myself why is the definition of this thing this way?
There's a saying in math: Analysts understand what they're talking about but find it hard to proving things - while algebraists don't know what they're talking about but find it easy to prove things.
I find this to be true, at least at the introductory level. Once you get to topology you forget what you're talking about, but that's the structural ("algebraic") view of math resurfacing. Maybe you're right though - perhaps there should be a progression in algebra from concrete -> abstract the same way there's a progression from concrete (real analysis) -> abstract (topology)
