Combinatorial conditions forcing commutativity of an infinite group

We show that the function f(n)=⌈(5n²-3n-2)/6⌉ 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| ≤ f(k), holds for every k-element subset K ⊆ G then G is abelian. Moreover, f(n) is the "maximal" integer valued function with this property. A characterization of central-by-finite groups appears in the proof.