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.

19 pages. v3: a mention to related independent work of L. Pyber has been included

References in corpus (2)

• Finite group actions on manifolds without odd cohomology
• Jordan groups and automorphism groups of algebraic varieties

Cited by in corpus (4)

• Finite group actions on manifolds without odd cohomology
• Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds
• Finite group actions on 4-manifolds with nonzero Euler characteristic
• On Jordan type bounds for finite groups of diffeomorphisms of 3-manifolds and Euclidean spaces