Combinatorial Conditions Forcing Commutativity of an Infinite Group Original Research Article

We show that the function ƒ(n)=left ceiling(5n2−3n−2)/6right ceiling is the best possible squaring bound for infinite abelian groups. That is, if G is an infinite group and k is an integer ≥ 2, such that the condition, K2 ≤ ƒ(k), holds for every k-element subset K subset of or equal to G then G is abelian. Moreover, ƒ(n) is the "maximal" integer valued function with this property. A characterization of central-by-finite groups appears in the proof.