A Sylow theorem for the integral group ring of PSL(2,q)

TL;DR

This paper proves a Sylow-type theorem for the units in the integral group ring of PSL(2,q), showing that units of prime power order are conjugate to group elements under certain conditions, extending understanding of unit structure.

Contribution

It establishes a Sylow theorem for units in the integral group ring of PSL(2,q), using the HeLP-method to connect units to group elements.

Findings

Units of r-power order are rationally conjugate to group elements.

Subgroups of prime power order in V(ZG) are conjugate to subgroups of G under certain conditions.

The result applies specifically when p=2 or f=1.

Abstract

For G = PSL(2,p^f) denote by ZG the integral group ring, by V(ZG) the group of normalized units of ZG and let r be a prime different from p. Using the so called HeLP-method we prove, that units of r-power order in V(ZG) are rationally conjugate to elements of G. As a consequence we prove, that subgroups of prime power order in V(ZG) are rationally conjugate to subgroups of G, provided p = 2 or f =1.