Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

Bächle, Andreas; Margolis, L.

doi:10.1017/s0013091516000535

Cited by 20 publications

(30 citation statements)

References 24 publications

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“…Remark 5.9. The HELP method can successfully be applied to the unique perfect group of order 120, SL (2,5). This proves the first Zassenhaus conjecture for SL(2, 5) × A, A a finite abelian group.…”

mentioning

confidence: 60%

“…(4) u is rationally conjugate to an element of G if and only if ε g (u d ) ≥ 0 for every g ∈ G and every d | n. (5) If N is a normal p-subgroup of G and u maps under the map ZG −→ ZG/N to 1, then u is a p-element.…”

Section: 1mentioning

confidence: 99%

“…Let 2a and 3a denote the unique A 5 -conjugacy class of involutions and elements of order 3, respectively. In all cases that cannot be ruled out, u 3 ∼ 2a, u 2 ∼ 3a and These cases can also not be ruled out by using Brauer characters (which might provide additional information in case of non-solvable groups) or the so-called lattice method [5].…”

Section: 4mentioning

confidence: 99%

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On the First Zassenhaus Conjecture and Direct Products

Bächle¹,

Kimmerle²,

Serrano³

2018

Can. J. Math.-J. Can. Math.

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In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products as well as the General Bovdi Problem (Gen-BP) which turns out to be a slightly weaker variant of (ZC1). Among others we prove that (Gen-BP) holds for Sylow tower groups, so in particular for the class of supersolvable groups.(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product G × A, where A is a finite abelian group and G has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group. /15. 1 As the name suggests, there are also a second and a third Zassenhaus conjecture dealing with conjugacy of (not necessarily cyclic) torsion subgroups of ZG. (See [36, Section 37] for more details.)

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confidence: 60%

Section: 1mentioning

confidence: 99%

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On the First Zassenhaus Conjecture and Direct Products

Bächle¹,

Kimmerle²,

Serrano³

2018

Can. J. Math.-J. Can. Math.

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“…Applying Proposition 5, we obtain: We shall now consider the prime graph of G = Aut(P SL(2, 13)). Clearly [2,3] and [2,7] are adjacent in π(G) and consequently adjacent in π(V (ZG)). However [2,13], [3,7], [3,13] and [7,13] are not adjacent in π(G).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…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].…”

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confidence: 97%

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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Algorithmic Aspects of Units in Group Rings

Bächle

Kimmerle

Margolis

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isomorphic to a subgroup of the group base. Moreover we study the orders of torsion units in integral group rings of finite almost quasisimple groups and the existence of torsion-free normal subgroups of finite index in the unit group.

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Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

Cited by 20 publications

References 24 publications

On the First Zassenhaus Conjecture and Direct Products

On the First Zassenhaus Conjecture and Direct Products

Torsion units for some almost simple groups

Algorithmic Aspects of Units in Group Rings

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