A variation of the Ito theorem (Q615878)

The authors show the following generalization of a famous result of Ito: If \(A\) and \(B\) are Abelian subgroups of the finite group \(G\) and the normal subset \(X\) of \(G\) is contained in \(AB\), then \(\langle X\rangle\) is metabelian. If further \(A\) and \(B\) are \(\pi\)-subgroups, so is \(\langle X\rangle\), if \(A\) and \(B\) are \(p\)-groups of exponent \(e\), then the exponent of \(\langle X\rangle\) is bounded by a function of \(e\).