There's certainly a question of appropriate pedagogy. In my own experience though, I find the abstract definitions easier to apply in novel situations rather than specific representations.

If I see a thing in the wild that has invertable operations, then I immediately think about an underlying group structure. As subgroups of Sn though, it's harder for me to make that connection.

When learning abstract algebra and category theory and such, thinking up several concrete examples on my own pretty much dissolves the "arbitrary list of axioms" feel for me and makes the definitions seem quite natural.