Boosting an analogue of Jordan's theorem for finite groups

Abstract: Let $\mathcal C$ be a set of finite groups which is closed under taking subgroups and let $d$ and $M$ be positive integers. 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. Our proofs use the Classification of Finite Simple Groups.