Finite Group Research Articles

Fractional revival on semi-Cayley graphs over abelian groups

In this paper, we investigate the existence of fractional revival on semi-Cayley graphs over finite abelian groups. We give some necessary and sufficient conditions for semi-Cayley graphs over finite abelian groups admitting fractional revival. We also show that integrality is necessary for some semi-Cayley graphs admitting fractional revival. Moreover, we characterize the minimum time when semi-Cayley graphs admit fractional revival. As applications, we give examples of certain Cayley graphs over the generalized dihedral groups and generalized dicyclic groups admitting fractional revival.

Lusztig correspondence and the finite Gan-Gross-Prasad problem

The Gan-Gross-Prasad problem is to describe the restriction of representations of a classical group G G to smaller groups H H of the same kind. In previous work by Liu and Wang [Trans. Amer. Math. Soc. 373 (2020), pp. 4223–4253; Manuscripta Math. 165 (2021), pp. 159–189; Math. Z. 297 (2021), pp. 997–1021] and Wang [Adv. Math. 393 (2021), p. 72], we study the Gan-Gross-Prasad problem for unipotent representations of finite classical groups. In this paper, we solved the Gan-Gross-Prasad problem over finite fields completely. The main tools used are the Lusztig correspondence and a formula of Reeder [J. Amer. Math. Soc. 20 (2007), pp. 573–602] for the pairings of Deligne-Lusztig characters. We give a reduction decomposition of Reeder’s formula and reduce the Gan-Gross-Prasad problem for arbitrary representations to the unipotent representations by Lusztig correspondence.

UNVEILING THE RELATIONSHIP BETWEEN M-POLYNOMIAL BASED TOPOLOGICAL INDICES AND INVERSE GRAPHS OF FINITE CYCLIC GROUPS: A COMPREHENSIVE STUDY

In the discipline of graph theory, topological indices are extremely important. The M-polynomial is a powerful tool for determining a graph's topological indices. The use of M-polynomials to describe macro-molecules and biochemical networking is a novel concept. Also, the M-polynomial of various micro-structural allows us to calculate a variety of topological indices. The chemical substances and biochemical networks are correlated with their chemical characteristics and bio-active compounds using these findings. In this research, we use the M-polynomial to create special essential topological indices of inverse graphs on finite cyclic groups, such as Randic, Zagreb, Augmented Zagreb, Harmonic, Inverse sum, and Symmetric division degree indices.

Superpower graphs of finite abelian groups

Superpower graphs of finite abelian groups

ON NEIGHBOURHOODS IN THE ENHANCED POWER GRAPH ASSOCIATED WITH A FINITE GROUP

Abstract We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

Integrability of generalised skew-symmetric replicator equations via graph embeddings

Abstract It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In this paper, we investigate the problem of determining whether these dynamics are Liouville-Arnold integrable, building on prior work graph in graph decloning by Evripidou et al. [J. Phys. A., 55:325201, 2022] and graph embedding by Paik and Griffin [Phys. Rev. E. 107(5): L052202, 2024]. Using the embedding procedure from Paik and Griffin, we show (with certain caveats) that when a graph producing integrable dynamics is embedded in another graph producing integrable dynamics, the resulting graph structure also produces integrable dynamics. We also construct a new family of graph structures that produces integrable dynamics that does not arise either from embeddings or decloning. We use these results, along with numerical methods, to classify the dynamics generated by almost all oriented directed graphs on six vertices, with three hold-out graphs that generate integrable dynamics and are not part of a natural taxonomy arising from known families and graph operations. These hold-out graphs suggest more structure is available to be found. Moreover, the work suggests that oriented directed graphs leading to integrable dynamics may be classifiable in an analogous way to the classification of finite simple groups, creating the possibility that there is a deep connection between integrable dynamics and combinatorial structures in graphs.

Computing Certain Properties of Additive and Multiplicative Groups of Integers Modulo n Utilizing MATLAB

This research focuses on two types of finite abelian groups: the group of integers under addition modulo n, and the group of integers under multiplication modulo n, where n is any positive integer up to 300. The computations in this study revolve around various properties of these groups, including the order of the group, the order and inverse of each element, the identification of cyclic subgroups, and the determination of generators within the group. To facilitate these calculations, a specialized program was developed using MATLAB. With this program, users can obtain answers for the aforementioned properties of these groups for any integer ranging from 0 to 300.

Units of twisted group rings and their correlations to classical group rings

This paper is centred around the classical problem of extracting properties of a finite group G from the ring isomorphism class of its integral group ring ZG. This problem is considered via describing the unit group U(ZG) generically for a finite group. Since the ‘90s’ several well known generic constructions of units are known to generate a subgroup of finite index in U(ZG) if QG does not have so-called exceptional simple epimorphic images, e.g. M2(Q). However it remained a major open problem to find a generic construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation U(ZG)ab. To obtain all this, we investigate the group ring RΓ of an extension Γ of some normal subgroup N by a group G, over a domain R. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of Γ over the fraction field F of R in terms of various twisted group rings of G over finite extensions of F. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of U(RΓ) via twisted group rings.

Powers in finite orthogonal and symplectic groups: A generating function approach

Powers in finite orthogonal and symplectic groups: A generating function approach

Representations of [formula omitted]-type combinatorial categories

In this paper we consider representations of certain combinatorial categories, including the poset D of positive integers and division, the Young lattice Y of partitions of finite sets, the opposite category of the orbit category Z of (Z,+) with respect to nontrivial subgroups, and the category CI of finite cyclic groups and injective homomorphisms. We describe explicit upper bounds for homological degrees of their representations, and deduce that finitely presented representations (resp., representations presented in finite degrees) over a field form abelian subcategories of the representation categories. We also give an explicit description for the category of sheaves over the ringed atomic site (Z,Jat,C_), and show that irreducible sheaves are parameterized by primitive roots of the unit.

Finitely generated normal pro-[formula omitted] subgroups in right angled Artin pro-[formula omitted] groups

Let C be a class of finite groups closed for subgroups, quotients groups and extensions. Let Γ be a finite simplicial graph and G=GΓ be the corresponding pro-C RAAG. We show that if N is a non-trivial finitely generated, normal, full pro-C subgroup of G then G/N is finite-by-abelian. In the pro-p case we show a criterion for N to be of type FPn when G/N≃Zp. Furthermore for G/N infinite abelian we show that N is finitely generated if and only if every normal closed subgroup N0◃G containing N with G/N0≃Zp is finitely generated. For G/N infinite abelian with N weakly discretely embedded in G we show that N is of type FPn if and only if every N0⩽G containing N with G/N0≃Zp is of type FPn.

A note on non-solvable groups with given number of particular subgroups

Considering the quantitative properties of some particular subgroups of a finite group, we prove that (1) a non-solvable group $G$ has exactly 5 non-subnormal non-supersolvable proper subgroups if and only if $G\cong A_5$ or $SL_2(5)$. (2) a non-solvable group $G$ has exactly 5 non-subnormal non-2-nilpotent proper subgroups if and only if $G\cong A_5$ or $SL_2(5)$. (3) a non-solvable group $G$ has exactly 16 non-subnormal non-2-closed proper subgroups (or two same order classes of non-subnormal non-2-closed proper subgroups) if and only if $G\cong A_5$ or $SL_2(5)$. Our results improve some known related research.

Local control and bogomolov multipliers of finite groups

ABSTRACT We show that if a Sylow p-subgroup of a finite group G is nilpotent of class at most p, then the p-part of the Bogomolov multiplier of G is locally controlled.

Adjacency Spectrum of Reduced Power Graphs of Finite Groups

Adjacency Spectrum of Reduced Power Graphs of Finite Groups

Holographic zero sound for [formula omitted] SCFTs in 4d

We build up the notion of metallic holography for N=2 SCFTs in four dimensions, in the presence of a finite U(1) chemical potential. We compute two point correlation functions and study their properties in the regime of low frequency and low momentum. The color sector reveals gapless excitations, one of which propagates with a finite phase velocity while the other mode attenuates through scattering. The flavour sector, on the other hand, reveals spectrum that contains quasiparticle excitations which propagate with a definite phase velocity together with modes that propagate with a finite group velocity which quantum attenuates quite similar in spirit as that of Landau's zero sound modes. We also compute associated AC conductivities, which exhibit different characteristics for different (color and flavour) sectors of N=2 SCFTs.

Observations about skew morphisms of cyclic groups

A permutation φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} on a finite group G is a skew morphism of G if φ(1G)=1G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (1_G)=1_G$$\\end{document} and there exists a function π:G→Z|⟨φ⟩|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi \\!: G \\rightarrow \\mathbb {Z}_{|\\langle \\varphi \\rangle |}$$\\end{document} such that φ(ab)=φ(a)φπ(a)(b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (ab) = \\varphi (a)\\varphi ^{\\pi (a)}(b)$$\\end{document} for all a,b∈G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a,b\\in G$$\\end{document}. In this paper, we develop an efficient algorithm for generating skew morphisms of finite cyclic groups and use it to determine all skew morphisms for cyclic groups up to order 1175. We then use this census to prove a number of new theorems about skew morphisms of finite cyclic groups. For example, we prove that the set of all skew morphisms of Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_n$$\\end{document} is itself a group if and only if all skew morphisms of Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_n$$\\end{document} are automorphisms, and we also determine all n for which every non-trivial skew morphism of Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_n$$\\end{document} gives rise to a regular Cayley map. In addition, we classify skew morphisms for all finite cyclic groups of orders p2q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^2q$$\\end{document} and p3q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^3q$$\\end{document} where p and q are primes satisfying p<q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<q$$\\end{document}.

Mini-Workshop: Growth and Expansion in Groups

The aim of the workshop was to give a panoramic view of the main lines of research on various aspects of growth and expansion in finite groups. The main topics included: diameter bounds for Cayley graphs of alternating and classical groups, with results for generating sets that can be arbitrary, random, or containing special elements; constructions of expander families of finite simple groups, with the use of property (T); and character bounds for finite simple groups, yielding applications to word map problems and random walks on Cayley graphs. A series of smaller topics explored connections and similarities to different problems in group theory.

Finite groups with many S-semipermutable p-subgroups

Finite groups with many S-semipermutable p-subgroups

A note on p-supersolubility of finite groups

A note on p-supersolubility of finite groups

On F^ω-projectors and F^ω-covering subgroups of finite groups

On F^ω-projectors and F^ω-covering subgroups of finite groups