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Suppose that for any $G\\in\\mathcal C$ whose order is divisible by at most two distinct primes there exists an abelian subgroup $A\\subseteq G$ such that $A$ is generated by at most $d$ elements and $[G : A] \\le M$. We prove that there exists a positive constant $C_0$ such that any $G \\in \\mathcal C$ has an abelian subgroup $A$ satisfying $[G : A] \\le C_0$, and $A$ can be generated by at most $d$ elements. We also prove some related results. 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