Lattice Of Subgroups Research Articles

Commensurators of normal subgroups of lattices

Commensurators of normal subgroups of lattices

Groups with finitely many isomorphism classes of non-modular subgroups

Groups in which the non-moduar subgroups fall into finitely many isomorphism classes are considered, and it is proved that a (generalized) soluble group with this property either has modular subgroup lattice or is a minimax group. The corresponding result for (generalized) soluble groups with finitely many isomorphism classes of non-permutable subgroups is also obtained.

Central products and the Chermak–Delgado lattice

The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups. We prove that the Chermak–Delgado lattice of a central product contains the product of the Chermak–Delgado lattices of the relevant central factors. Furthermore, we obtain information about heights of elements in the Chermak–Delgado lattice relative to their heights in the Chermak–Delgado lattices of central factors. We also explore how the central product can be used as a tool in investigating Chermak–Delgado lattices.

Groups and their coset monoids

This paper studies groups and their coset monoids. The anti-abnormal subgroups of a group are firstly introduced and investigated. It is shown that a group is an N ˜ -group if and only if each subgroup of it is anti-abnormal. Also, it is proved that the coset semigroups of N ˜ -groups are exactly the E-reflexive inverse semigroups which are factorisable and the natural connection between their semilattice of idempotents and lattice of subgroups of their group of units is a dual isomorphism. Finally, some characterizations of the coset semigroups of S-groups (U-groups and residually central groups respectively) are given. This extends McAlister’s result in 1980.

On the lattice of closed subgroups of a profinite group

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behavior of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.

Möbius function of the subgroup lattice of a finite group and Euler characteristic

The Möbius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the Möbius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup G of the general linear group GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}(n,q)$$\\end{document} acting on the n-dimensional vector space V=Fqn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V=\\mathbb {F}_q^n$$\\end{document}, where Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_q$$\\end{document} is the finite field with q elements. We find a relation between this function and the Euler characteristic of two simplicial complexes Δ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _1$$\\end{document} and Δ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _2$$\\end{document}, the former raising from the lattice of the subspaces of V, the latter from the subgroup lattice of G.

Counting and Equidistribution of Reciprocal Geodesics and Dihedral Groups

Abstract We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in $\operatorname{PSL}_{2}{\mathbb{R}}$, generalizing the earlier work of Sarnak [ 9] and Bourgain–Kontorovich [ 4] on the growth of the number of reciprocal geodesics on the modular surface. We also prove that reciprocal geodesics are equidistributed in the unit tangent bundle.

Limiting Distribution of Dense Orbits in a Moduli Space of Rank m Discrete Subgroups in (m+1)-Space

Abstract We study the limiting distribution of dense orbits of a lattice subgroup $\Gamma \leq \textrm{SL}(m+1,\mathbb{R})$ acting on $H\backslash \textrm{SL}(m+1,\mathbb{R})$, with respect to a filtration of growing norm balls. The novelty of our work is that the groups $H$ we consider have infinitely many non-trivial connected components. For a specific such $H$, the homogeneous space $H\backslash G$ identifies with $X_{m,m+1}$, a moduli space of rank $m$-discrete subgroups in $\mathbb{R}^{m+1}$. This study is motivated by the work of Shapira-Sargent who studied random walks on $X_{2,3}$.

Large norms in group theory

In 1935, the introduction of the norm of a group by Reinhold Baer is a turning point in group theory. In fact, Baer proved that there is a very strong relationship between the structure of the norm and that of the whole group (see [1], [2], [3], [4], [5]). Since then, the norm has been playing a very significant roles in many aspects of group theory and its applications: it has been used in [43] to describe the connection between Hopf–Galois structures and skew braces; it has been used in [23] to describe some special types of profinite groups; and it has been fundamental in the theory of subgroup lattices of groups (see [40]).In this paper, we weaken the original definition of norm by taking into account only those subgroups that are “large” in some sense. Depending on the chosen concept of largeness, the resulting norm can have an impact on the structure of the whole group that is even greater than that of Baer's norm. This is exactly what happens with the non-polycyclic norm, and in fact, Theorem 4.17 gives a precise description of generalized soluble groups in which the non-polycyclic norm is non-Dedekind (and can be considered as the main result of the paper). Other times, the resulting norms have their own peculiar behaviour; this is the case if “large” means “infinite”, “having infinite rank”, “being non-Černikov”, or “having cardinality m” for some given uncountable cardinal number m.

An investigation on a fewproperties of the subgroup lattice of 2 × 2 matrices over z11

An investigation on a fewproperties of the subgroup lattice of 2 × 2 matrices over z11

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

We prove that in a cocompact complex hyperbolic arithmetic lattice Γ<PU(m,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Gamma < {\\mathrm{PU}}(m,1)$\\end{document} of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type Fm−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathscr{F}_{m-1}$\\end{document} but not of type Fm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathscr{F}_{m}$\\end{document}. This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.

A closure operator on the subgroup lattice of GL(𝑛,𝑞) and PGL(𝑛,𝑞) in relation to the zeros of the Möbius function

Abstract Let F q \mathbb{F}_{q} be the finite field with 𝑞 elements and consider the 𝑛-dimensional F q \mathbb{F}_{q} -vector space V = F q n V=\mathbb{F}_{q}^{n} . In this paper, we define a closure operator on the subgroup lattice of the group G = PGL ⁢ ( V ) G=\mathrm{PGL}(V) . Let 𝜇 denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups 𝐻 of 𝐺 for which μ ⁢ ( H , G ) ≠ 0 \mu(H,G)\neq 0 . Moreover, we establish a polynomial bound on the number c ⁢ ( m ) c(m) of closed subgroups 𝐻 of index 𝑚 in 𝐺 for which the lattice of 𝐻-invariant subspaces of 𝑉 is isomorphic to a product of chains. This bound depends only on 𝑚 and not on the choice of 𝑛 and 𝑞. It is achieved by considering a similar closure operator for the subgroup lattice of GL ⁢ ( V ) \mathrm{GL}(V) and the same results proven for this group.

Aggregation of T-subgroups of groups whose subgroup lattice is a chain

In this work, we study when an aggregation operator preserves the structure of T-subgroup of groups whose subgroup lattice is a chain. There are two widely used ways of defining the aggregation of structures in fuzzy logic, previously named on sets and on products. We will focus our attention on the one called aggregation on products. When the lattice of subgroups is not a chain, it is known that the dominance relation between the aggregation operator and the t-norm is crucial. We show that this property is again important for some of the groups in this study. However, for the rest of them, we must define a new property weaker than domination, that will allow us to characterize those operators which preserve T-subgroups.

On some new developments in the theory of subgroup lattices of groups

On some new developments in the theory of subgroup lattices of groups

LATTICE UNIVERSALITY OF LOCALLY FINITE \(p\)-GROUPS

For an arbitrary prime \(p\), we prove that every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of a suitable locally finite \(p\)-group. In particular, every lattice is embeddable in the subgroup lattice of a locally finite \(p\)-group.

Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows

We show an analog of a theorem of [J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of [Formula: see text], Adv. Math. 354 (2019) 106743] on weighted badly approximable vectors for totally imaginary number fields. We show that for [Formula: see text] and [Formula: see text] a lattice subgroup, the points of [Formula: see text] with bounded orbits under a one-parameter Ad-semisimple subgroup of [Formula: see text] form a hyperplane-absolute-winning set. As an application, we also provide a generalization of a result of [R. Esdahl-Schou and S. Kristensen, On badly approximable complex numbers, Glasgow Math. J. 52 (2010) 349–355] about the set of badly approximable complex numbers.

ON THE NUMBER OF FUZZY SUBGROUPS AND FUZZY NORMAL SUBGROUPS OF S2, S3 AND A4

Counting fuzzy subgroups of a finite group is a fundamental problem of fuzzy group theory. Many researchers have made significant contributions to the rapid growth of this topic in recent years. The number of fuzzy subgroups of any group is infinite without the aid of equivalence relation. Some authors have used the equivalence relation of fuzzy sets to study the equivalence of fuzzy subgroups ([5], [6], [16]). The problem of counting the number of distinct fuzzy subgroups of a finite group is relative to the choice of the equivalence relation. The number of fuzzy subgroups of a particular group varies from one equivalence relation to the other. The equivalence relation applied in our computation can be seen in the existing literature. Sulaiman and Abd Ghafur [10] define an equivalence relation for counting fuzzy subgroups of group G. We have used this relation to find fuzzy subgroups and fuzzy normal subgroups of S2, S3 and A4. Lattice subgroup diagrams were used in our computation.

ARITHMETICITY OF -FUCHSIAN SUBGROUPS OF SOME NONARITHMETIC LATTICES

AbstractWe study the arithmeticity of $\mathbb {C}$ -Fuchsian subgroups of some nonarithmetic lattices constructed by Deraux et al. [‘New non-arithmetic complex hyperbolic lattices’, Invent. Math.203 (2016), 681–771]. Our results give an answer to a question raised by Wells [Hybrid Subgroups of Complex Hyperbolic Isometries, Doctoral thesis, Arizona State University, 2019].

Congruence Lattices of Ideals in Categories and (Partial) Semigroups

This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.