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A generalization of the Brauer-Fowler theorem

Saveliy V. Skresanov

TL;DR

The paper generalizes the Brauer–Fowler theorem to simple locally finite groups by proving that if a simple locally finite group contains an involution $t$ that commutes with at most $n$ involutions, then the group is finite and its order is bounded in terms of $n$ only. For finite simple groups, the proof combines a Guralnick–Robinson bound on the number of involution conjugacy classes with a rank bound for Lie-type groups and a fixed-point/character ratio analysis (Gluck bounds) to deduce $|G| \le C_1 \cdot n^{C_2 n^2}$ with universal constants $C_1,C_2>0$. If there is only one conjugacy class of involutions, the argument splits by the parity of $q$: for even $q$ one shows $q \le n+1$; for odd $q$ a finite list of possibilities is reduced by embedding $\mathrm{PSL}_2(q)$ into larger Lie-type groups to bound $q$. Extending the result to simple locally finite groups uses a Kegel cover to pass to finite quotients, and the finite-quotient bound yields finiteness of $G$; the paper also discusses open problems about periodic simple groups.

Abstract

The famous Brauer-Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most $n$ involutions, then the group is finite and its order is bounded in terms of $n$ only. This answers a question of Strunkov from the Kourovka notebook.

A generalization of the Brauer-Fowler theorem

TL;DR

The paper generalizes the Brauer–Fowler theorem to simple locally finite groups by proving that if a simple locally finite group contains an involution that commutes with at most involutions, then the group is finite and its order is bounded in terms of only. For finite simple groups, the proof combines a Guralnick–Robinson bound on the number of involution conjugacy classes with a rank bound for Lie-type groups and a fixed-point/character ratio analysis (Gluck bounds) to deduce with universal constants . If there is only one conjugacy class of involutions, the argument splits by the parity of : for even one shows ; for odd a finite list of possibilities is reduced by embedding into larger Lie-type groups to bound . Extending the result to simple locally finite groups uses a Kegel cover to pass to finite quotients, and the finite-quotient bound yields finiteness of ; the paper also discusses open problems about periodic simple groups.

Abstract

The famous Brauer-Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most involutions, then the group is finite and its order is bounded in terms of only. This answers a question of Strunkov from the Kourovka notebook.
Paper Structure (1 section, 3 theorems, 4 equations)

This paper contains 1 section, 3 theorems, 4 equations.

Table of Contents

  1. Acknowledgements

Key Result

Proposition 1

Let $G$ be a periodic simple group, and let $t, z \in G$ be nonconjugate involutions. If the sets $C_G(t) \cap (t^G \cup z^G)$ and $C_G(z) \cap (t^G \cup z^G)$ are finite and have at most $n$ elements, then $G$ is also finite and its order can be bounded in terms of $n$ only.

Theorems & Definitions (3)

  • Proposition 1: S
  • Theorem 1: mod CFSG
  • Proposition 2: GRcomm