Combinatorial conditions forcing commutativity of an infinite group (Q1327045)

The author defines the integer function \(f\) by \(f(k)=(5k^2-3k)/6\) for \(3\mid k\) and \(f(k)=(5k^2-2k-2)/6\) otherwise, and then shows that if every \(k\)-element subset \(K\) of a group has \(|K^2|\leq f(k)\), then the group is Abelian; and, moreover, the function \(f\) is best possible, in the sense for \(k\geq 2\) there exists a strictly non-Abelian group such that \(|K^2|\leq f(k)+1\) for every \(k\)-element subset \(K\) of the group. On the way to the proof he proves the following lemma, which is of independent interest: The centre of a group has finite index in it [that is to say, the group is a so-called FIZ-group] if, and only if, there is an integer \(k\geq 2\) such that every \(k\)-element subset \(K\) of the group satisfies the inequality \(|K^2|\leq k^2-k\).