I prefer thinking about group theory as the study of the symmetric group of order n. Then to make sense of the case of n = infinity, you introduce the group axi

I prefer thinking about group theory as the study of the symmetric group of order n. Then to make sense of the case of n = infinity, you introduce the group axioms. Eventually, you can justify the study of permutations using the fact that if you take any algebraic structure and consider the group of automorphisms, you get a group.