Bass units as free factors in integral group rings of simple groups

TL;DR

This paper investigates Bass units in integral group rings of simple groups, classifies relevant groups, and proves a conjecture about free groups generated by certain units for specific cases, supported by computational verification.

Contribution

It classifies simple groups with dihedral p-critical elements and proves the conjecture for several cases, advancing understanding of Bass units in group rings.

Findings

Classified simple groups with dihedral p-critical elements as PSL(2,q).

Proved the conjecture for p=5, even q, and q+1=2p cases.

Verified the conjecture for all q<10000 with computational methods.

Abstract

Let $G$ be a finite group, $u$ a Bass unit based on an element $a$ of $G$ of prime order, and assume that $u$ has infinite order modulo the center of the units of the integral group ring $\Z G$. It was recently proved that if $G$ is solvable then there is a Bass unit or a bicyclic unit $v$ and a positive integer $n$ such that the group generated by $u^n$ and $v^n$ is a non-abelian free group. It has been conjectured that this holds for arbitrary groups $G$. To prove this conjecture it is enough to do it under the assumption that $G$ is simple and $a$ is a dihedral $p$-critical element in $G$. We first classify the simple groups with a dihedral $p$-critical element. They are all of the form $\PSL(2,q)$. We prove the conjecture for $p=5$; for $p>5$ and $q$ even; and for $p>5$ and $q+1=2p$. We also provide a sufficient condition for the conjecture to hold for $p>5$ and $q$ odd. With the help of computers we have verified the sufficient condition for all $q<10000$.