On the Zassenhaus conjecture for PSL(2,q), SL(2,q) and direct products

Abstract

In 1960 Zassenhaus proposed several conjectures about the finite subgroups of U(ZG) where G is a finite group. Counterexamples for most of them appeared during the following years but one of them was opened for a long time. It was the conjecture which pretend to describe the torsion elements of U(ZG). Clearly, the conjugates in QG of the elements of G and their opposite have finite order and the Zassenhaus Conjecture (ZC1) predicts that every torsion element of ZG is of this form. (ZC1) has been proved for groups of order at most 144, for nilpotent groups by Weiss and for metacyclic groups by Hertweck. This result was extended to cyclic-by-abelian groups. Before finishing this thesis, a metabelian counterexample to (ZC1) was announced by Eisele and Margolis. As a consequence, weaker versions are gaining importance. We point out the Kimmerle Problem which tries to decide whether the torsion units of ZG are conjugate to elements of G in U(QH) for a group H containing G as subgroup. The goals of this thesis have been: 1) Study (ZC1) for the groups PSL(2,q) and SL(2,q). 2) Develop new methods to deal with (ZC1). 3) Study (ZC1) and the Kimmerle Problem for direct products. To reach goals 1) and 2), examples proved using only the HeLP Method were studied. In this thesis it has been calculated how far one can go after applying the HeLP Method for units of order 2t in ZPLS(2,q), with t an odd prime. It is also proved that every torsion unit of ZPSL(2,q) of order coprime with 2q is conjugate in QG to an element of PSL(2,q). For this, a new version of the HeLP Method suitable to the characters of PSL(2,q) was developed. As a consequence, it is proved (ZC1) for the groups PSL(2,p) with p a Fermat or Mersenne prime. This result increases the number of non-abelian simple groups for which (ZC1) holds from 13 to at least 62 groups. It is studied whether the techniques used for the groups PSL(2,q) also work to deal with (ZC1) for the groups SL(2,q). For that, the Modular Representation Theory of these groups was deeply studied with the hope that it could help to deal with larger classes of groups. In this thesis it is proved that every torsion unit of ZSL(2,q) with order coprime to q is conjugate in QG to an element of SL(2,q). This yields to the proof of (ZC1) for the groups SL(2,p) and SL(2,p²) with p a prime. This is the first infinite family of non-solvable groups for which (ZC1) has been proved. Regarding goal 3), it is studied whether the techniques used by Hertweck could be adapted to deal with direct products. As a consequence, it is proved (ZC1) for the direct product of an abelian finite group and a finite Frobenius group with metacylic complements. The classical HeLP Method was also extended to group rings where the coefficients are coming from ring of algebraic integers in order to study (ZC1) for the direct product of an abelian finite group and a group whose order is at most 95. In this thesis, the Kimmerle Problem was solved for the counterexamples to (ZC1) obtained by Eisele and Margolis and for groups with a Sylow tower, in particular for supersolvable groups.