Zassenhaus conjecture on torsion units holds for SL(2, 𝑝) and SL(2, 𝑝2)

Abstract

Abstract H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring ℤ ⁢ G {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra ℚ ⁢ G {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups SL ⁢ ( 2 , p ) {\mathrm{SL}(2,p)} and SL ⁢ ( 2 , p 2 ) {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if G = SL ⁢ ( 2 , p f ) {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of ℤ ⁢ G {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in ℚ ⁢ G {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of ℤ ⁢ G {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.