Near subgroups of groups (Q1903720)

The authors define a near subgroup of a group \(G\) as a finite subset \(A\) with the property that one has \(xy \notin A\) for at most \(|A |\) pairs \((x,y) \in A^2\). Natural examples arise by taking \(A\) as a subgroup \(H\) of \(G\) from which at most one element has been deleted. The authors prove that these are the only examples of near subgroups provided that \(|A|\geq 5\). They also classify the counterexamples for \(|A|\leq 4\).