Groups containing locally maximal product-free sets of size 4

Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand sets, Bertram asked whether every abelian satisfy |S−−√|≤2|S|. This question was recently answered the negation by current author. Here, we improve some results on structures sizes of groups terms their sets. A consequence our is classification that contain sets size 4, continuing work Street, Whitehead, Giudici Hart containing small sizes. We also obtain partial arbitrary conclude with conjecture 4 problem as well an open general case.