Integral Group Ring of the Mathieu Simple GroupM₂₃

Abstract

Abstract. We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M 24 . As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.

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

“…It was also confirmed for some Sporadic Simple groups in [21][22][23][24][25][26][27][28][29][30][31][32]. Additionally, it was confirmed for the sympletic simple group S 4 (4) in [33].…”

“…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.…”

Integral group ring of the Mathieu simple groupM 12

“…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.…”

“…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.…”

Torsion units for some almost simple groups

“…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.…”