|
Let \(G\) be a finite group and \(V(\mathbb{Z}G)\) the group of normalized units in the intgeral group ring of \(G.\) A long standing conjecture of H.J. Zassenhaus asks, whether for every torsion unit \(u \in V(\mathbb{Z}G)\) there exists a unit in \(\mathbb{Q}G\) conjugating \(u\) onto an element of \(G.\) In case there is, one says, that \(u\) is rationally conjugate to a group element. A weaker form of the Zassenhaus Conjecture, the so called Primegraph Question, asks, whether it follows from \(V(\mathbb{Z}G)\) having an element of order \(pq,\) that \(G\) also posseses an element of order \(pq,\) for every pair of different prime numbers \(p\) and \(q\).
Several results concerning this questions for \(G=\operatorname{PSL}(2,p^f)\) will be presented in this talk. Especially the result, that every torsion subgroup of prime power order of \(V(\mathbb{Z}\operatorname{PSL}(2,p^f))\) is rationally conjugate to a subgroup of \(G\) provided \(f \leq 2\) or \(p=2.\) The methods involved are the so called HeLP-method and a method developed by the speaker and A. B??chle from an argument of M. Hertweck, which involves integral and modular representation theory.
|
