A new solvability criterion for finite groups

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In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if for all conjugacy classes C and D of G, there exist x∈ C and y∈ D for which x,y is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two integers a and b which represent orders of elements in G and for all elements x,y∈ G with |x|=a and |y|=b, the subgroup x,y is nonsolvable.