Abstract. We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M 12 . As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.
Introduction, conjectures and main resultsLet V (ZG) be the normalized unit group of the integral group ring ZG of a finite group G. A long-standing conjecture of H. Zassenhaus (ZC) says that every torsion unit u ∈ V (ZG) is conjugate within the rational group algebra QG to an element in G.For finite simple groups the main tool for the investigation of the Zassenhaus conjecture is the Luthar-Passi method, introduced in [17] to solve it for A 5 . Later M. Hertweck in [14] extended the Luthar-Passi method and applied it for the investigation of the Zassenhaus conjecture for P SL(2, p n ). The Luthar-Passi method proved to be useful for groups containing non-trivial normal subgroups as well. For some recent results we refer to [5,7,12,14,13,15]. Also, some related properties and some weakened variations of the Zassenhaus conjecture can be found in [1,18] and [3,16].First of all, we need to introduce some notation. By #(G) we denote the set of all primes dividing the order of G. The Gruenberg-Kegel graph (or the prime graph) of G is the graph π(G) with vertices labeled by the primes in #(G) and with an edge from p to q if there is an element of order pq in the group G. In [16] W. Kimmerle proposed the following weakened variation of the Zassenhaus conjecture:(KC) If G is a finite group then π(G) = π(V (ZG)). In particular, in the same paper W. Kimmerle verified that (KC) holds for finite Frobenius and solvable groups. Note that with respect to the so-called p-version of the Zassenhaus conjecture the investigation of Frobenius groups was completed by M. Hertweck and the first author in [4]. In [6,7,8] (KC) was confirmed for sporadic simple groups M 11 , M 23 and some Janko simple groups.Here we continue these investigations for the Mathieu simple group M 12 . Although using the Luthar-Passi method we cannot prove the rational conjugacy for torsion units of V (ZM 12 ), our main result gives a lot of information on partial 1991 Mathematics Subject Classification. Primary 16S34, 20C05, secondary 20D08.