Commutativity theorems for groups are statements with the following structure: If a group G satisfies property P, then G is commutative. For example, if in a group G every element is the inverse of itself, then G is commutative. In 2016, G. Venkataraman, an expert in groups holding a PhD from Oxford, proved the following commutativity theorem: If in a finite group the squares commute and the cubes commute, then the group is commutative. She also offered the conjecture that the result holds for arbitrary groups. Our main result is the following. Let S be a subsemigroup of a samilattice of cancellative semigroups. If the p-powers commute and the q-powers commute, for p and q natural coprime numbers, then S is commutative. This result, in particular, fully answers Venkataraman´s question.


Francisco Miguel Araújo