Abstract
Let C be a set of finite groups which is closed under taking subgroups and let d and M be positive integers. Suppose that for every G∈C whose order is divisible by at most two distinct primes there exists an abelian subgroup A⊆G such that A is generated by d or fewer elements and [G:A]≤M. We prove that there exists a positive constant C0 such that every G∈C has an abelian subgroup A satisfying [G:A]≤C0, and A can be generated by d or fewer elements. We also prove some related results. Our proofs use the Classification of Finite Simple Groups.
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