Group Theory and Related Topics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 18548

Special Issue Editors

Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100 Burjassot, València, Spain
Interests: abstract group theory; finite groups; infinite groups; braces; Yang-Baxter equation; automata theory; formal language theory
Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100 Burjassot, València, Spain
Interests: abstract group theory; finite groups; infinite groups; braces; Yang-Baxter equation; automata theory; formal language theory

Special Issue Information

Dear Colleagues,

The study of algebraic equations; the analysis of the symmetry, or, in general, the transformations that respect certain geometric structures; and number theory are three of the branches of mathematics that led at the end of the 19th century to the formalization of the notion of group. Group theory has become one of the core disciplines of algebra, and many algebraic structures can be regarded as “groups with an additional structure.”

Some relevant results on groups were obtained during the second half of the 20th century.  One of them was the completion of the classification of finite simple groups in 1992. The theory of classes of groups provides a language to describe group-theoretical properties that has been recently exported to other algebraic structures and has attracted the interest of not only mathematicians but also computer scientists. These two milestones and other important results on group theory have opened new research lines with regard to this theory.

Since the generalisation of the use of computers, algorithmic methods in group theory have been an active source of research, and computer algebra systems have become fundamental tools for current research on groups, especially in the finite case. Groups also lie on the foundations of modern cryptography. The complexity of some problems in groups are a source of cryptographic algorithms. The fact that some groups act on other geometric or algebraic structures has been crucial to finding new applications of group theory, not only to algebra but also to fields related to geometry, topology, physics, and chemistry.

This aim of this Special Issue of Mathematics is to show recent results on group theory and other related topics. We cordially invite you to present your recent contributions to this Special Issue.

Yours sincerely,

Prof. Adolfo Ballester-Bolinches
Prof. Ramón Esteban-Romero
Guest Editors

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Keywords

  • Group theory
  • Finite group
  • Infinite group
  • Action of groups
  • Representation theory
  • Geometric group theory
  • Topological group theory
  • Asymptotic group theory
  • Generalisations of groups
  • Applications of groups.

Published Papers (9 papers)

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Research

25 pages, 402 KiB  
Article
Bounds on the Number of Maximal Subgroups of Finite Groups: Applications
by Adolfo Ballester-Bolinches, Ramón Esteban-Romero and Paz Jiménez-Seral
Mathematics 2022, 10(7), 1153; https://doi.org/10.3390/math10071153 - 02 Apr 2022
Cited by 3 | Viewed by 1922
Abstract
The determination of bounds for the number of maximal subgroups of a given index in a finite group is relevant to estimate the number of random elements needed to generate a group with a given probability. In this paper, we obtain new bounds [...] Read more.
The determination of bounds for the number of maximal subgroups of a given index in a finite group is relevant to estimate the number of random elements needed to generate a group with a given probability. In this paper, we obtain new bounds for the number of maximal subgroups of a given index in a finite group and we pin-point the universal constants that appear in some results in the literature related to the number of maximal subgroups of a finite group with a given index. This allows us to compare properly our bounds with some of the known bounds. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
7 pages, 241 KiB  
Article
On Groups in Which Many Automorphisms Are Cyclic
by Mattia Brescia and Alessio Russo
Mathematics 2022, 10(2), 262; https://doi.org/10.3390/math10020262 - 15 Jan 2022
Viewed by 1252
Abstract
Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup x,xα is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: [...] Read more.
Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup x,xα is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: On a class of normal endomorphisms of groups, J. Algebra and its Applications 13, (2014), 6pp] the authors proved that every cyclic automorphism is central, namely, that every cyclic automorphism acts trivially on the factor group G/Z(G). In this paper, the class FW of groups in which every element induces by conjugation a cyclic automorphism on a (normal) subgroup of finite index will be investigated. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
17 pages, 328 KiB  
Article
A Compendium of Infinite Group Theory: Part 1—Countable Recognizability
by Francesco de Giovanni and Marco Trombetti
Mathematics 2021, 9(19), 2366; https://doi.org/10.3390/math9192366 - 24 Sep 2021
Cited by 2 | Viewed by 1742
Abstract
Countably recognizable group classes were introduced by Reinhold Baer and provide a very ingenious way to study large groups through the properties of their countable subgroups. This is the reason we have chosen the countable recognizability to start this series of survey papers [...] Read more.
Countably recognizable group classes were introduced by Reinhold Baer and provide a very ingenious way to study large groups through the properties of their countable subgroups. This is the reason we have chosen the countable recognizability to start this series of survey papers on infinite group theory. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
11 pages, 313 KiB  
Article
The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph
by Siqiang Yang and Xianhua Li
Mathematics 2021, 9(17), 2060; https://doi.org/10.3390/math9172060 - 26 Aug 2021
Viewed by 1242
Abstract
Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. [...] Read more.
Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. We obtain a series of theorems which generalize some existed results. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
8 pages, 259 KiB  
Article
Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds
by Raeyong Kim
Mathematics 2021, 9(12), 1330; https://doi.org/10.3390/math9121330 - 09 Jun 2021
Viewed by 1399
Abstract
The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a [...] Read more.
The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
6 pages, 258 KiB  
Article
On the Norm of the Abelian p-Group-Residuals
by Baojun Li, Yu Han, Lü Gong and Tong Jiang
Mathematics 2021, 9(8), 842; https://doi.org/10.3390/math9080842 - 13 Apr 2021
Cited by 1 | Viewed by 1240
Abstract
Let G be a group. Dp(G)=HGNG(H(p)) is defined and, the properties of Dp(G) are investigated. It is proved that [...] Read more.
Let G be a group. Dp(G)=HGNG(H(p)) is defined and, the properties of Dp(G) are investigated. It is proved that Dp(G)=P[A], where P=D(P) is the Sylow p-subgroup and A=N(A) is a Hall p-subgroup of Dp(G), respectively. Furthermore, it is proved in a group G that (1) Dp(G)=1 if and only if CG(G(p))=1; (2) Op(Dp(G))Z(Op(G)) and (3) if Z(G(p))=1, then CG(G(p))=Dp(G). Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
4 pages, 235 KiB  
Article
On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups
by Abd El-Rahman Heliel, Mohammed Al-Shomrani and Adolfo Ballester-Bolinches
Mathematics 2020, 8(12), 2165; https://doi.org/10.3390/math8122165 - 04 Dec 2020
Cited by 6 | Viewed by 1513
Abstract
Let σ={σi:iI} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of [...] Read more.
Let σ={σi:iI} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)i for some i{0,1,2}. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
8 pages, 258 KiB  
Article
Products of Finite Connected Subgroups
by María Pilar Gállego, Peter Hauck, Lev S. Kazarin, Ana Martínez-Pastor and María Dolores Pérez-Ramos
Mathematics 2020, 8(9), 1498; https://doi.org/10.3390/math8091498 - 04 Sep 2020
Viewed by 1516
Abstract
For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if a,bL for all aA and [...] Read more.
For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if a,bL for all aA and bB. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
19 pages, 514 KiB  
Article
The Derived Subgroups of Sylow 2-Subgroups of the Alternating Group, Commutator Width of Wreath Product of Groups
by Ruslan V. Skuratovskii
Mathematics 2020, 8(4), 472; https://doi.org/10.3390/math8040472 - 30 Mar 2020
Cited by 10 | Viewed by 1992
Abstract
The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group A 2 k is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator [...] Read more.
The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group A 2 k is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of A 2 k is constructed. It is shown that ( S y l 2 A 2 k ) 2 = S y l 2 A 2 k , k > 2 . It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups C p i , p i N equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width ( c w ( G ) ) of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups S y l 2 ( A 2 k ) of A 2 k is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group A 2 k , where k > 2 , the permutation group S 2 k , as well as Sylow p-subgroups of S y l 2 A p k as well as S y l 2 S p k ) are equal to 1 was obtained. A commutator width of permutational wreath product B C n is investigated. An upper bound of the commutator width of permutational wreath product B C n for an arbitrary group B is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP. Full article
(This article belongs to the Special Issue Group Theory and Related Topics)
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