For finite G=AB, the condition that <a,b> is soluble for all a in A and b in B is equivalent to the commutator [A,B] lying in the soluble radical of G.
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Thompson-like characterization of solubility for products of finite groups
For finite G=AB, the condition that <a,b> is soluble for all a in A and b in B is equivalent to the commutator [A,B] lying in the soluble radical of G.