Torsion Units in Integral Group Ring of Higman-sims Simple Group

Let V (ZG) be the normalized unit group of the integral group ring ZG of a finite group G. One of most interesting conjectures in the theory of integral group ring is the conjecture (ZC) of H. Zassenhaus [25], saying that every torsion unit u ∈ V (ZG) is conjugate to an element in G within the rational group algebra QG. For finite simple groups, the main tool of the investigation of the Zassenhaus conjecture is the Luthar–Passi method, introduced in [21] to solve the (ZC) for A5. Later in [16] M. Hertweck extended and applied it for the investigation of the Zassenhaus conjecture for PSL(2, p). The method proved to be useful for groups containing non-trivial normal subgroups as well. We refer to [5, 7, 15, 16, 17, 18] for recent results. Related results can be found in [1, 22] and [3, 20]. In the latter papers weakened versions of the (ZC) were conjectured. In order to state one of these we 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 labelled by the primes in #(G) and there is an edge from p to q if and only if there is an element of order pq in the group G. In [20] W. Kimmerle proposed the following: Conjecture (KC): if G is a finite group then π(G) = π(V (ZG)). Obviously, the Zassenhaus conjecture (ZC) implies the Kimmerle conjecture (KC). In [20] it was shown, that (KC) holds for finite Frobenius and solvable groups. We remark 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 [7, 8, 9, 11], (KC) was also confirmed for certain Mathieu sporadic simple groups, and in [6] – for some Janko sporadic simple groups. In this paper we continue these investigations for the Higman-Sims simple sporadic group HS. The main result provides information about the possible torsion units in V (ZHS). An immediate consequence is a positive answer to (KC) for HS.