Boosting an analogue of Jordan's theorem for finite groups

TL;DR

This paper extends a Jordan-type theorem for finite groups, establishing uniform bounds on abelian subgroups in classes of groups with restricted prime divisors, using the Classification of Finite Simple Groups.

Contribution

It proves a uniform bound on the index of abelian subgroups in certain classes of finite groups, generalizing previous results and employing the Classification of Finite Simple Groups.

Findings

Existence of a universal constant C_0 for abelian subgroup index

Bounded generation of abelian subgroups by at most d elements

Application of Classification of Finite Simple Groups in proofs

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.