[lmm]

You need to be able to write the group axioms down, so you need to distinguish the three elements. E.g. one of the axioms is 1 ∈ G, which means something very different from G ∈ 1. It all bottoms out in set theory, but (G, 1, X) really is a different set from (G, X, 1) (e.g. they might be represented as the sets {{G, 1, X}, {G, 1}, {G}} and {{G, 1, X}, {G, X}, {G}} respectively) so you do need to decide which is the "canonical" representation of that group, or else you need a rule that allows you to tell whether two different sets are representations of the same group.