Published by De Gruyter

Volume 24 Issue 4

Issue of Journal of Group Theory

Contents
Journal Overview

Publicly Available July 1, 2021

Frontmatter

Page range: i-iv

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Publicly Available January 20, 2021

Permutation groups with small orbit growth

Manuel Bodirsky, Bertalan Bodor

Page range: 643-709

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Abstract

Let 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} be the class of all structures 𝔄{\mathfrak{A}} such that the automorphism group of 𝔄{\mathfrak{A}} has at most c⁢nd⁢n{cn^{dn}} orbits in its componentwise action on the set of n -tuples with pairwise distinct entries, for some constants c,d{c,d} with d<1{d<1}. We show that 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of finite covers of first-order reducts of unary structures, and also that 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}}. We also show that Thomas’ conjecture holds for 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}}: all structures in 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} have finitely many first-order reducts up to first-order interdefinability.

January 5, 2020

Graphical splittings of Artin kernels

Enrique Miguel Barquinero, Lorenzo Ruffoni, Kaidi Ye

Page range: 711-735

Abstract

We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.

February 2, 2021

Detecting conjugacy stability of subgroups in certain classes of groups

Isabel Fernández Martínez, Denis Serbin

Page range: 737-751

Abstract

In this paper, we consider the conjugacy stability property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free hyperbolic groups and finitely generated subgroups of limit groups.

February 16, 2021

Condensed groups in product varieties

Denis Osin

Page range: 753-763

Abstract

A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U⁢V\mathcal{UV}, where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in U⁢V\mathcal{UV}.

January 12, 2021

New Lie products for groups and their automorphisms

James B. Wilson

Page range: 765-780

Abstract

We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these rings to the automorphisms of a group. This process uncovers new structure in 4/5 of the approximately 11.8 million groups of size at most 1000 and beyond that point pertains to at least a positive logarithmic proportion of all finite groups.

Publicly Available February 2, 2021

Quasi-powerful 𝑝-groups

James Williams

Page range: 781-806

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Abstract

In this paper, we introduce the notion of a quasi-powerful p -group for odd primes p . These are the finite p -groups G such that G/Z⁢(G){G/Z(G)} is powerful in the sense of Lubotzky and Mann. We show that this large family of groups shares many of the same properties as powerful p -groups. For example, we show that they have a regular power structure, and we generalise a result of Fernández-Alcober on the order of commutators in powerful p -groups to this larger family of groups. We also obtain a bound on the number of generators of a subgroup of a quasi-powerful p -group, expressed in terms of the number of generators of the group, and we give an example which demonstrates this bound is close to best possible.

December 23, 2020

Subnormality and residuals for saturated formations: A generalization of Schenkman’s theorem

Stefanos Aivazidis, Inna N. Safonova, Alexander N. Skiba

Page range: 807-818

Abstract

Let G be a finite group, and let 𝔉{\mathfrak{F}} be a hereditary saturated formation. We denote by 𝐙𝔉⁢(G){\mathbf{Z}_{\mathfrak{F}}(G)} the product of all normal subgroups N of G such that every chief factor H/K{H/K} of G below N is 𝔉{\mathfrak{F}}-central in G , that is, (H/K)⋊(G/𝐂G⁢(H/K))∈𝔉{(H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F}}. A subgroup A⩽G{A\leqslant G} is said to be 𝔉{\mathfrak{F}}-subnormal in the sense of Kegel , or K -𝔉{\mathfrak{F}}-subnormal in G , if there is a subgroup chain A=A0⩽A1⩽⋯⩽An=G{A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=G} such that either Ai-1⁢⊴⁢Ai{A_{i-1}\trianglelefteq A_{i}} or Ai/(Ai-1)Ai∈𝔉{A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F}} for all i=1,…,n{i=1,\ldots,n}. In this paper, we prove the following generalization of Schenkman’s theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let F{\mathfrak{F}} be a hereditary saturated formation containing all nilpotent groups, and let S be a K -F{\mathfrak{F}}-subnormal subgroup of G . If ZF⁢(E)=1{\mathbf{Z}_{\mathfrak{F}}(E)=1} for every subgroup E of G such that S⩽E{S\leqslant E}, then CG⁢(D)⩽D{\mathbf{C}_{G}(D)\leqslant D}, where D=SF{D=S^{\mathfrak{F}}} is the F{\mathfrak{F}}-residual of S .

October 22, 2020

Groups in which each subnormal subgroup is commensurable with some normal subgroup

Ulderico Dardano, Fausto De Mari

Page range: 819-834

Abstract

We study groups in which each subnormal subgroup is commensurable with a normal subgroup. Recall that two subgroups 𝐻 and 𝐾 are termed commensurable if H∩KH\cap K has finite index in both 𝐻 and 𝐾. Among other results, we show that if a (sub)soluble group 𝐺 has the above property, then 𝐺 is finite-by-metabelian, i.e., G′′G^{\prime\prime} is finite.

Open Access January 14, 2021

Odd order products of conjugate involutions in linear groups over GF(2^𝑎)

John J. Ballantyne, Peter J. Rowley

Page range: 835-856

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Abstract

Let G be isomorphic to GLn⁢(q){\mathrm{GL}_{n}(q)}, SLn⁢(q){\mathrm{SL}_{n}(q)}, PGLn⁢(q){\mathrm{PGL}_{n}(q)} or PSLn⁢(q){\mathrm{PSL}_{n}(q)}, where q=2a{q=2^{a}}. If t is an involution lying in a G -conjugacy class X , then, for arbitrary n , we show that, as q becomes large, the proportion of elements of X which have odd order product with t tends to 1. Furthermore, for n at most 4, we give formulae for the number of elements in X which have odd order product with t , in terms of q .

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

Volume 24 Issue 4

Issue of Journal of Group Theory

Frontmatter

Permutation groups with small orbit growth

Abstract

Graphical splittings of Artin kernels

Abstract

Detecting conjugacy stability of subgroups in certain classes of groups

Abstract

Condensed groups in product varieties

Abstract

New Lie products for groups and their automorphisms

Abstract

Quasi-powerful 𝑝-groups

Abstract

Subnormality and residuals for saturated formations: A generalization of Schenkman’s theorem

Abstract

Groups in which each subnormal subgroup is commensurable with some normal subgroup

Abstract

Odd order products of conjugate involutions in linear groups over GF(2𝑎)

Abstract

Odd order products of conjugate involutions in linear groups over GF(2^𝑎)