group axioms

The group axioms are the four-item checklist that decides whether a set with an operation deserves the name “group.” Think of them as a quality-control inspection: any candidate must pass all four to be admitted. They are short enough to memorize, yet strong enough that the entire theory of groups is squeezed out of them.

The four are: (1) Closure — for all a, b in G, a * b is also in G. (2) Associativity — (a * b) * c = a * (b * c) for all a, b, c. (3) Identity — there is an element e with e * a = a * e = a for every a. (4) Inverses — for each a there is an element b with a * b = b * a = e. (Closure is sometimes folded into the very definition of the operation, leaving three axioms; either way the requirements are the same.)

Two famous consequences come essentially for free: the identity is unique, and each element's inverse is unique. What is deliberately absent is commutativity (a * b = b * a) — that is not required, and a group with it is given the extra title abelian. Checking these four boxes is the standard first step whenever you meet a new candidate structure.