Published by De Gruyter

Volume 24 Issue 1

Issue of Journal of Group Theory

Contents
Journal Overview

Publicly Available January 1, 2021

Frontmatter

Page range: i-iv

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Publicly Available August 25, 2020

A periodicity theorem for acylindrically hyperbolic groups

Oleg Bogopolski

Page range: 1-15

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Abstract

We generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

Open Access August 25, 2020

Right-angled Artin groups and enhanced Koszul properties

Alberto Cassella, Claudio Quadrelli

Page range: 17-38

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Abstract

Let 𝔽{\mathbb{F}} be a finite field. We prove that the cohomology algebra H∙⁢(GΓ,𝔽){H^{\bullet}(G_{\Gamma},\mathbb{F})} with coefficients in 𝔽{\mathbb{F}} of a right-angled Artin group GΓ{G_{\Gamma}} is a strongly Koszul algebra for every finite graph Γ. Moreover, H∙⁢(GΓ,𝔽){H^{\bullet}(G_{\Gamma},\mathbb{F})} is a universally Koszul algebra if, and only if, the graph Γ associated to the group GΓ{G_{\Gamma}} has the diagonal property. From this, we obtain several new examples of pro- p groups, for a prime number p , whose continuous cochain cohomology algebra with coefficients in the field of p elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro- p Galois groups of fields formulated by J. Mináč et al.

August 25, 2020

Irreducible representations of Braid group 𝐵_𝑛 of dimension 𝑛 + 1

Inna Sysoeva

Page range: 39-78

Abstract

We prove that there are no irreducible representations of Bn{B_{n}} of dimension n+1{n+1} for n⩾10{n\geqslant 10}.

Open Access August 15, 2020

Invariable generation and wreath products

Charles Garnet Cox

Page range: 79-93

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Abstract

Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various authors have investigated the property for particular infinite groups or families of infinite groups. A group is invariably generated by a subset S if replacing each element of S with any of its conjugates still results in a generating set for G . In this paper, we investigate how this property behaves with respect to wreath products. Our main work is to deal with the case where the base of G≀XH{G\wr_{X}H} is not invariably generated. We see both positive and negative results here depending on H and its action on X .

August 25, 2020

Groups satisfying the double chain condition on non-subnormal subgroups

Mattia Brescia

Page range: 95-108

Abstract

If θ is a subgroup property, a group G is said to satisfy the double chain condition on θ-subgroups if it admits no infinite double sequences ⋯<X-n<⋯<X-1<X0<X1<⋯<Xn<⋯\cdots<X_{-n}<\cdots<X_{-1}<X_{0}<X_{1}<\cdots<X_{n}<\cdots consisting of θ-subgroups. The structure of generalised radical groups satisfying the double chain condition on non-subnormal subgroups is described.

Publicly Available August 12, 2020

The main decomposition of finite-dimensional protori

Wayne Lewis, Peter Loth, Adolf Mader

Page range: 109-127

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Abstract

A protorus is a compact connected abelian group of finite dimension. We use a result on finite-rank torsion-free abelian groups and Pontryagin duality to considerably generalize a well-known factorization of a finite-dimensional protorus into a product of a torus and a torus free complementary factor. We also classify direct products of protori of dimension 1 by means of canonical “type” subgroups. In addition, we produce the duals of some fundamental theorems of discrete abelian groups.

August 19, 2020

Local finiteness of the twisted Bruhat orders on affine Weyl groups

Weijia Wang

Page range: 129-160

Abstract

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra 163 1994, 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set B in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed B -twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.

Open Access August 25, 2020

Generation of finite groups with cyclic Sylow subgroups

Heiko Dietrich, Darren Low

Page range: 161-175

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Abstract

M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. We generalise Slattery’s result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.

September 9, 2020

Finite 𝑝-groups all of whose 𝓐₂-subgroups are generated by two elements

Lihua Zhang, Junqiang Zhang

Page range: 177-193

Abstract

Assume G is a finite p -group. We prove that if all 𝒜2{\mathcal{A}_{2}}-subgroups of G are generated by two elements, then so are all non-abelian subgroups of G . By using this result, we classify the p -groups which have at least two 𝒜1{\mathcal{A}_{1}}-subgroups and in which the intersection of every pair of distinct 𝒜1{\mathcal{A}_{1}}-subgroups equals the intersection of all the 𝒜1{\mathcal{A}_{1}}-subgroups. It turns out that such p -groups are the finite p -groups all of whose non-abelian subgroups are generated by two elements, and the converse also almost holds.

September 9, 2020

Even character degrees and normal Sylow 2-subgroups

Hongfei Pan, Nguyen Ngoc Hung, Shuqin Dong

Page range: 195-205

Abstract

The Ito–Michler theorem on character degrees states that if a prime p does not divide the degree of any irreducible character of a finite group G , then G has a normal Sylow p -subgroup. We give some strengthened versions of this result for p=2{p=2} by considering linear characters and those of even degree.

About this journal

Objective

The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

Topics

Group Theory
Representation Theory of Groups
Computational Aspects of Group Theory
Combinatorics and Graph Theory
Algebra and Number Theory

Article formats

Original research articles

Information on submission process

Volume 24 Issue 1

Issue of Journal of Group Theory

Frontmatter

A periodicity theorem for acylindrically hyperbolic groups

Abstract

Right-angled Artin groups and enhanced Koszul properties

Abstract

Irreducible representations of Braid group 𝐵𝑛 of dimension 𝑛 + 1

Abstract

Invariable generation and wreath products

Abstract

Groups satisfying the double chain condition on non-subnormal subgroups

Abstract

The main decomposition of finite-dimensional protori

Abstract

Local finiteness of the twisted Bruhat orders on affine Weyl groups

Abstract

Generation of finite groups with cyclic Sylow subgroups

Abstract

Finite 𝑝-groups all of whose 𝓐2-subgroups are generated by two elements

Abstract

Even character degrees and normal Sylow 2-subgroups

Abstract

Irreducible representations of Braid group 𝐵_𝑛 of dimension 𝑛 + 1

Finite 𝑝-groups all of whose 𝓐₂-subgroups are generated by two elements