A set G together with a binary operation for which the following properties are satisfied:

(1) Associativity: (ab)c = a(bc) for all a, b, c in G.

(2) Identity: there exists an element e in G for which ae = ea = a for all a in G.

(3) Inverses: For each element a in G, there is an element b in G for which ab = ba = e.