Representation theory of finite groups I: A primer on group theory

Why these axioms? The symmetries of a given object form a group under composition, and the axioms capture the general properties of such a composition law. Any object has at least one symmetry, namely the identity transformation, which does nothing at all. Any symmetry of a given object can be undone or reversed, whence the existence of inverses. Associativity is the most subtle condition: it says that the composition of several symmetries (in a given order) is well-defined regardless of how we evaluate the composition. Said more formally, we can move parentheses around freely, so in particular the expression x1 · · ·xn makes sense for a finite collection x1, · · · , xn ∈ G. A fourth axiom which might seem natural is commutativity, which says that any x, y ∈ G commute, meaning xy = yx. Groups which have this property are called abelian. Abelian groups are in many ways easier to understand than nonabelian groups.