Global and local properties of finite groups with only finitely many central units in their integral group ring

Bächle, Andreas; Caicedo, Mauricio; Jespers, Eric; Maheshwary, Sugandha

doi:10.1515/jgth-2020-0165

Cited by 4 publications

(10 citation statements)

References 30 publications

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“…Now consider an element 2(𝑥 + 𝑥) + 6 with |𝑥| 2 = 2. Then, using (R1), the fact that modulo E 2 () ′ all elements commute, that 𝐸(0) 4…”

Section: 3mentioning

confidence: 99%

“…For example, rational groups are cut. Recently, cut groups gained in interest (see, e.g., [4, 7, 68]), but especially the subclass of rational groups has already a long tradition in classical representation theory (e.g., see [37, 50]). Theorem Let G be a finite group.…”

Section: Introductionmentioning

confidence: 99%

“…the group  (ℤ𝐺) has property HFA, 2. the group  (ℤ𝐺) has property HFℝ, 3. the group  (ℤ𝐺) has property (T), 4. the group  (ℤ𝐺) has property FAb, 5.…”

Section: Introductionmentioning

confidence: 99%

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Abelianization and fixed point properties of units in integral group rings

Bächle

Janssens

Jespers

et al. 2022

Mathematische Nachrichten

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Let 𝐺 be a finite group and  (ℤ𝐺) the unit group of the integral group ring ℤ𝐺. We prove a unit theorem, namely, a characterization of when  (ℤ𝐺) satisfies Kazhdan's property (T), both in terms of the finite group 𝐺 and in terms of the simple components of the semisimple algebra ℚ𝐺. Furthermore, it is shown that for  (ℤ𝐺), this property is equivalent to the weaker property FAb (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in  (ℤ𝐺) have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SL 𝑛 (), where  is an order in a finite-dimensional semisimple ℚ-algebra 𝐷, and finite groups 𝐺, which have the so-called cut property. For such groups 𝐺, we describe the simple epimorphic images of ℚ𝐺. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups E 𝑛 (𝐷) of SL 𝑛 (𝐷). These groups are well understood except in the degenerate case of lower rank, that is, for SL 2 () with  an order in a division algebra 𝐷 with a finite number of units. In this setting, we determine Serre's property FA for E 2 () and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ℤ-rank.

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“…Now consider an element 2(𝑥 + 𝑥) + 6 with |𝑥| 2 = 2. Then, using (R1), the fact that modulo E 2 () ′ all elements commute, that 𝐸(0) 4…”

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Abelianization and fixed point properties of units in integral group rings

Bächle

Janssens

Jespers

et al. 2022

Mathematische Nachrichten

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“…The class of cut groups contains the class of finite rational groups but is considerably bigger, e.g. 86.62% of all groups up to order 512 are cut while only 0.57% of them are rational [BCJM21,Section 7]. Since it turned out that G being a cut group is a major obstruction for certain fixed point properties (such as Kazhdan's property (T)) of the unit group of its integral group ring ZG, this class of groups also appeared naturally in the study of these properties and in the proof of a virtual unit theorem for non-trivial amalgamated products [BJJ + 21, BJJ + 18].…”

Section: Introductionmentioning

confidence: 99%

Gruenberg-Kegel graphs: cut groups, rational groups and the Prime Graph Question

Bächle¹,

Kiefer²,

Maheshwary³

et al. 2021

Preprint

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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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“…[Seh93], Corollary 1.7). Further, a group G such that all central units in ZG are trivial, is termed as cut-group; this class of groups is currently a topic of active research ([BMP17], [Mah18], [Bäc18], [BCJM18], [BMP19], [Tre19]). In this article, we quite often use the characterization of abelian cut-groups, namely, an abelian group G is a cut-group, if and only if its exponent divides 4 or 6.…”

Section: Introductionmentioning

confidence: 99%