Zassenhaus Conjecture on torsion units holds for $\mathrm{PSL}(2,p)$ with $p$ a Fermat or Mersenne prime

Abstract

H.J. Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of the form $\pm g$ with $g\in G$. Though known for some series of solvable groups, the conjecture has been proved only for thirteen non-abelian simple groups. We prove the Zassenhaus Conjecture for the groups $\mathrm{PSL}(2,p)$, where $p$ is a Fermat or Mersenne prime. This increases the list of non-abelian simple groups for which the conjecture is known by probably infinitely many, but at least by 49, groups. Our result is an easy consequence of known results and our main theorem which states that the Zassenhaus Conjecture holds for a unit in $\mathbb{Z}\mathrm{PSL}(2,q)$ of order coprime with $2q$, for some prime power $q$.

13 pages, a much shorter version. arXiv admin note: text overlap with arXiv:1803.05342

References in corpus (3)

• Partial augmentations and Brauer character values of torsion units in group rings
• On the Prime Graph Question for Integral Group Rings of 4-primary groups II
• On the torsion units of the integral group ring of finite projective special linear groups

Cited by in corpus (2)

• On the torsion units of the integral group ring of finite projective special linear groups
• Algorithmic aspects of units in group rings