Zassenhaus conjecture for central extensions of S 5

Abstract

We confirm a conjecture of Zassenhaus about rational conjugacy of torsion units in integral group rings for a covering group of the symmetric group S 5 and for the general linear group GLð2; 5Þ. The first result, together with others from the literature, settles the conjugacy question for units of prime-power order in the integral group ring of a finite Frobenius group.

Integral Group Ring of the Mathieu Simple GroupM23

“…We remark that with respect to (ZC) the investigation of Frobenius groups was completed by M. Hertweck and the first author in [4]. In [6,7,8,9, 11] (KC) was confirmed for the Mathieu simple groups M 11 , M 12 , M 22 , M 23 and the sporadic Janko simple groups J 1 , J 2 and J 3 .…”

Torsion units for a Ree group, Tits group and a Steinberg triality group

“…The Zassenhaus conjecture was confirmed true for all groups up to order 71 in [6]. This conjecture was also validated for A 5 , S 5 , central extensions of S 5 and other simple finite groups in [5,[7][8][9]. In [10] partial results were given for A 6 , and the remaining cases were dealt with in [11].…”

“…In order to prove that the Zassenhaus Conjecture holds, we need to consider torsion units of V (ZG) of order 2,3,4,6,7,8,9,12,13,14,18,21,28,24,26,36,39,42,56,63 and 91 (by Proposition 1). For the purpose of this paper and due to the complexity of certain orders, we shall consider elements of order 2, 3, 7, 13, 26, 39 and 91.…”

“…G contains elements of order 6, 14 and 21. Therefore [2,3], [2,7] and [3,7] are adjacent in π(G) and consequently adjacent in π(V (ZG)). Clearly π(G) = π(V (ZG)), since there are no torsion units of order 26, 39 and 91 in V (ZG).…”

Integral group ring of the Mathieu simple groupM 12

“…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.…”