Finite groups with many product conjugacy classes. (Q2472708)

Let \(K\) and \(L\) be two conjugacy classes of a group \(G\). Then the product \(KL\) will contain the identity element of \(G\) if and only there exists \(x\in K\) with \(x^{-1}\in L\). Also observe that in this case \(KL\) is a conjugacy class if and only if \(K=\{x\}\) and \(L=\{x^{-1}\}\). In this case we can define \(L=K^{-1}\). This paper classifies those finite groups \(G\) such that \(KL\) is a conjugacy class whenever \(K\neq L^{-1}\). They prove that in this case there are only four possibilities: (a) \(G\) is a finite Abelian group; (b) \(G\) is a non-Abelian Camina \(p\)-group; (c) \(\text{AGL}(1,F)\) where \(F\) is a finite field with at least \(3\) elements; and (d) the split extension of an elementary Abelian group of order \(9\) with the quaternion group of order \(8\). If the condition is weakened that \(KL\) is a conjugacy class whenever \(KZ(G)\neq L^{-1}Z(G)\) they prove that the group is isoclinic to a group satisfying the stronger condition. The proof is divided into three sections. The first proves some basic facts and shows that the group is soluble. The second looks at nilpotent groups and the third completes the proof.