Finite simple groups with short Galois orbits on conjugacy classes

Bovdi, Victor; Breuer, Thomas; Maróti, Attila

doi:10.1016/j.jalgebra.2019.10.024

Cited by 4 publications

(1 citation statement)

References 19 publications

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“…Consequently, it follows that if V has finite abelianization, then the center of V consists only of trivial units and in this case, G is known as a cut groupcentral units trivial [BMP17]. This class of groups has been intensively studied lately [BMP17,Mah18,Bäc18,BCJM18,Tre19,BBM20]. From yet another perspective, understanding V /V ′ can also be regarded as a first important step towards understanding the lower central series of V , an important object for which very few results are available.…”

Section: Introductionmentioning

confidence: 99%

Abelianization of the unit group of an integral group ring

Bächle,

Maheshwary,

Margolis

2020

Preprint

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For a finite group G and U := U(ZG), the group of units of the integral group ring of G, we study the implications of the structure of G on the abelianization U/U ′ of U . We pose questions on the connections between the exponent of G/G ′ and the exponent of U/U ′ as well as between the ranks of the torsion-free parts of Z(U ), the center of U , and U/U ′ . We show that the units originating from known generic constructions of units in ZG are well-behaved under the projection from U to U/U ′ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ + 18].The following variations of (2) appear natural for V : (R1) Is rank V /V ′ = rank Z(V )? (R2) Assume Z(V ) is finite. Is V /V ′ also finite?The above questions address the free subgroup of V /V ′ , while the ones listed next try to get hold of its torsion elements. Denote by exp H, the exponent of a group H, i.e., the greatest common divisor of the orders of torsion elements of H.

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Section: Introductionmentioning

confidence: 99%

Abelianization of the unit group of an integral group ring

Bächle,

Maheshwary,

Margolis

2020

Preprint

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On K _n -simple groups and traces of maximal subgroups

Huang,

Xiang,

Zhang

et al. 2024

Communications in Algebra

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Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question

Bächle¹,

Kiefer

Maheshwary

et al. 2023

Forum Mathematicum

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The Gruenberg–Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices p, q are joined by an edge whenever the group has an element of order pq. It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well-studied class of rational groups, and has received attention recently. In the first part of this paper, we give a complete classification of the Gruenberg–Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg–Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg–Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg–Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups for the classes of cut groups and rational groups, respectively. The prime graph question asks whether the Gruenberg–Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the first Zassenhaus conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the prime graph question for integral group rings for finite rational groups and most finite cut groups

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Finite simple groups with short Galois orbits on conjugacy classes

Cited by 4 publications

References 19 publications

Abelianization of the unit group of an integral group ring

Abelianization of the unit group of an integral group ring

On K _n -simple groups and traces of maximal subgroups

Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question

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Finite simple groups with short Galois orbits on conjugacy classes

Cited by 4 publications

References 19 publications

Abelianization of the unit group of an integral group ring

Abelianization of the unit group of an integral group ring

On K n -simple groups and traces of maximal subgroups

Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question

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Product

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On K _n -simple groups and traces of maximal subgroups