Kimmerle conjecture for the Held and O'Nan sporadic simple groups

Abstract

Using the Luthar-Passi method, we investigate the Zassenhaus and Kim- merle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held sim- ple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture. Let U(ZG) be the unit group of the integral group ring ZG of a finite group G. It is well known that U(ZG )= U(Z) × V (ZG), where V (ZG )= g∈G αgg ∈ U(ZG) | g∈G αg =1 ,α g ∈ Z.