Torsion units for some almost simple groups

Abstract

Abstract. We prove that the Zassenhaus conjecture is true for Aut(P SL(2, 11)). Additionally we prove that the Prime graph question is true for Aut(P SL(2, 13)). Introduction and main resultsLet U (ZG) be the unit group of the integral group ring of a finite group G. It is well known that U (ZG) = {±1} × V (ZG), where V (ZG) is the group of units of augmentation one. Throughout this article, G is always a finite group and torsion units will always represent torsion units in V (ZG) \ {1}. A very important conjecture in the theory of integral group rings is: Conjecture 1. If G is a finite group, then for each torsion unit u ∈ V (ZG) there exists g ∈ G, such that |u| = |g| where |u| and |g| is the order of u and g respectively.A stronger version of this conjecture was formulated by Hans Zassenhaus in [37], which states that: Conjecture 2. A torsion unit in V (ZG) is said to be rationally conjugate to a group element if it is conjugate to an element of G by a unit of the rational group algebra QG.This conjecture was confirmed for some classes of solvable groups in [23], nilpotent groups in [32,36] and cyclic-by-abelian groups in [18]. The Luthar-Passi Method (which was introduced in [29]) is the main investigative tool for simple groups G in relation to the Zassenhaus conjecture for ZG. It was confirmed true for all groups up to order 71, A 5 , S 5 , central extensions of S 5 and other simple finite groups in [4,5,25,29,30]. Partial results were given for A 6 in [33] and the remaining cases were dealt with in [21]. Higher order alternating groups were also considered in [34,35]. It was also proved for P SL(2, p) when p = {7, 11, 13} in [22], P SL(2, p) when p = {8, 17} in [19] and P SL(2, p) when p = {19, 23} in [2]. Further results regarding P SL(2, p) can be found in [24].Let H be a group with a torsion part t(H) (i.e. the set of elements of H of finite order) of finite exponent and #H be the set of primes dividing the order of elements from the set t(H). The prime graph of H (denoted by π(H)) is a graph with vertices labeled by primes from #H, such that vertices p and q are adjacent if and only if there is an element of order pq in the group H. The following, was composed as a problem in [31] (Problem 37):This question was upheld for Frobenius and Solvable groups in [27] and was also confirmed for some Sporadic Simple groups in [3,[6][7][8][9][10][11][12][13][14][15][16]. We use the Luthar-Passi Method to obtain our results. Our results are the following: Theorem 1. The Zassenhaus Conjecture is true for the integral group ring of the automorphism group of the group P SL(2, 11).