Finite groups of units and their composition factors in the integral group rings of the groups PSL(2, q)

Hertweck, Martin; Höfert, Christian R.; Kimmerle, Wolfgang

doi:10.1515/jgt.2009.019

Cited by 13 publications

(13 citation statements)

References 22 publications

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Section: Introduction and Main Resultsmentioning

confidence: 72%

“…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).…”

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confidence: 71%

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Torsion units for some almost simple groups

Gildea

2016

Czech Math J

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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).

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Section: Introduction and Main Resultsmentioning

confidence: 72%

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confidence: 71%

Torsion units for some almost simple groups

Gildea

2016

Czech Math J

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“…Quite satisfactory results have been obtained for the series PSL(2, q) [14]. In this article we continue this study with respect to the series of the Suzuki groups Sz(q) = 2 B 2 (q), q = 2 2m+1 , in order to get results for all minimal simple groups G.…”

Section: Introductionmentioning

confidence: 87%

On torsion subgroups in integral group rings of finite groups

Bächle

Kimmerle

2011

Journal of Algebra

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The object of this article are torsion subgroups of the normalized unit group V(ZG) of the integral group ring ZG of a finite group G. For specific subgroups W we study the GruenbergKegel graph Π(W ). It is shown that the central elements of an isolated subgroup U of a group basis H of ZG are the normalized units of its centralizer ring C ZG (U ). Moreover Π(N V(ZG) (U )) = Π (N H (U )). If G has elementary abelian Sylow 2-subgroups of order at most 8 each finite 2-subgroup of V(ZG) is rationally conjugate to a subgroup of G. Finally torsion subgroups of V(ZG) in the case when G is a minimal simple group are considered. It follows that if G is a simple group which admits a non-trivial partition for each prime p the p-rank of a torsion subgroup of V(ZG) is bounded by that one of G.

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“…In [10] it is proved that the r-subgroups of V(ZPSL(2, p f )) are isomorphic to subgroups of PSL(2, p f ) for all primes r provided p = 2 or f ≤ 2. In the concluding remarks, the question whether also for f ≥ 3 all p-subgroups of V(ZPSL(2, p f )) are (elementary) abelian is highlighted.…”

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confidence: 99%