Lattice Of Subgroups Research Articles

The lattice of closure operators on a subgroup lattice

ABSTRACTWe say a lattice L is a subgroup lattice if there exists a group G such that Sub(G)≅L, where Sub(G) is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of closure operators which act on the subgroup lattice of a finite group G is itself a subgroup lattice if and only if G is cyclic of prime power order.

Lattice extensions of Hecke algebras

We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma–Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a structure theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the general case. We prove that these algebras are symmetric algebras at least when W is a Coxeter group, and in general under the trace conjecture of Broué, Malle and Michel.

Finding intermediate subgroups

This article describes a practical approach for determining the lattice of subgroups U < V < G between given subgroups U and G , provided the total number of such subgroups is not too large. It builds on existing functionality for element conjugacy, double cosets and maximal subgroups.

The Chermak–Delgado measure in finite p-groups

Let G be a finite group and let H≤G. The Chermak–Delgado measure ([7]) of H is defined as the number |H‖CG(H)|. The subgroups of G having maximum measure, the F1(G)-subgroups, form a sublattice F1(G) in the lattice of subgroups of G. The maximum measure equals |K‖Z(K)| for some subgroup K. In [8], for p-groups, subgroups satisfying this property are called centrally large.We continue here the study of F1-subgroups and centrally large subgroups of p-groups. Then we focus on self minimal centrally largep-groups (SMCL), i.e. p-groups having no proper centrally large subgroups. We characterize SMCL-p-groups having proper non central F1-subgroups as central products, and then we study central products of SMCL-p-groups. Finally, we exhibit some classes of p-groups that are not SMCL.

Ore’s theorem on cyclic subfactor planar algebras and beyond

Ore proved that a finite group is cyclic if and only if its subgroup lattice is distributive. Now, since every subgroup of a cyclic group is normal, we call a subfactor planar algebra cyclic if all its biprojections are normal and form a distributive lattice. The main result generalizes one side of Ore's theorem and shows that a cyclic subfactor is singly generated in the sense that there is a minimal 2-box projection generating the identity biprojection. We conjecture that this result holds without assuming the biprojections to be normal, and we show that it is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a non-trivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.

On the table of marks of a direct product of finite groups

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group S3.

On the Structure of the Schild Group in Relativity Theory

AbstractAlfred Schild has established conditions that Lorentz transformationsmap world-vectors (ct, x, y, z) with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in SL(z, ℂ). We consider the lattice of subgroups of the group originated in Schild’s paper and obtain generating sets for the full group and its subgroups.

Finite groups with a trivial Chermak–Delgado subgroup

AbstractThe Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups ofG. The least element of the Chermak–Delgado lattice ofGis known as the Chermak–Delgado subgroup ofG. This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichainp-group lattices play a major role in the author’s constructions.

Cellular automata and finite groups

For a finite group $G$ and a finite set $A$, we study various algebraic aspects of cellular automata over the configuration space $A^G$. In this situation, the set $\text{CA}(G;A)$ of all cellular automata over $A^G$ is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units $\text{ICA}(G;A)$ of $\text{CA}(G;A)$. We obtain a decomposition of $\text{ICA}(G;A)$ into a direct product of wreath products of groups that depends on the numbers $\alpha_{[H]}$ of periodic configurations for conjugacy classes $[H]$ of subgroups of $G$. We show how the numbers $\alpha_{[H]}$ may be computed using the M\"obius function of the subgroup lattice of $G$, and we use this to improve the lower bound recently found by Gao, Jackson and Seward on the number of aperiodic configurations of $A^G$. Furthermore, we study generating sets of $\text{CA}(G;A)$; in particular, we prove that $\text{CA}(G;A)$ cannot be generated by cellular automata with small memory set, and, when all subgroups of $G$ are normal, we determine the relative rank of $\text{ICA}(G;A)$ on $\text{CA}(G;A)$, i.e. the minimal size of a set $V \subseteq \text{CA}(G;A)$ such that $\text{CA}(G;A) = \langle \text{ICA}(G;A) \cup V \rangle$.

Frankl's Conjecture for Subgroup Lattices

We show that the subgroup lattice of any finite group satisfies Frankl’s Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.

The Krull-Schmidt Theorem Holds for Finite Direct Products of Biuniform Groups

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups $\mathcal {N}(G)$ has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups.

Maximal torsion-free subgroups of certain lattices of hyperbolic buildings and Davis complexes

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.

On finite groups for which the lattice of S-permutable subgroups is distributive

A subgroup A of a finite group G is said to permute with a subgroup B if \(AB=BA\). If A permutes with all Sylow subgroups of G, then A is called S-permutable in G. We characterize finite groups with modular and distributive lattice of S-permutable subgroups.

Free 3-Generated Lattices with Two Semi-Normal Generators

We consider a lattice generated by three elements, two of which are semi-normal. We have proved it is infinite and have also found an additional condition for the lattice to be modular. As a result, it is proved that the sublattice generated in the subgroup lattice by two normal subgroups and any subgroup is modular.

A broad class of shellable lattices

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k-equal partition lattices.We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To begin with, the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable. We thus obtain a unified proof that lattices in these classes are shellable.Subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable. We show not only that subgroup lattices of solvable groups are comodernistic, but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice. Indeed, the definition of comodernistic exactly requires on every interval a lattice-theoretic analogue of the composition series in a solvable group. Thus, the relation between comodernistic lattices and solvable groups resembles, in several respects, that between supersolvable lattices and supersolvable groups.

The Homomorphism Lattice Induced by a Finite Algebra

Each finite algebra A induces a lattice L A via the quasi-order → on the finite members of the variety generated by A, where B →C if there exists a homomorphism from B to C. In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice L A induced by a finite algebra A?’ Our main result is that each finite distributive lattice arises as L Q , for some quasi-primal algebra Q. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form L ⊕1, where L is an interval in the subgroup lattice of a finite group.

Fundamental isomorphism theorems for quantum groups

The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into “chunks” (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus’ butterfly lemma, the Schreier refinement theorem, the Dedekind modularity law, and last but not least the Jordan–Hölder theorem.We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latter Hopf algebraic, hence the corresponding two-part organization of our study. Our intention is that the analytic portion be accessible to the algebraist and vice versa.The upshot is that in the locally compact case one often needs further assumptions (integrability, compactness, discreteness). In the linearly reductive case on the other hand, the quantum versions of the results hold without further assumptions. Moreover the case of compact/discrete quantum groups is usually covered by both the linearly reductive and the locally compact framework, thus providing a bridge between the two.

The annihilation graphs of commutator posets and lattices with respect to an element

We propose a new, widely generalized context for the study of the zero-divisor type (annihilating-ideal) graphs, where the vertices of graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set [lattice] (imitating the lattice of ideals of a ring), equipped with a commutative (not necessarily associative) binary operation (imitating the product of ideals of a ring). We discuss, when [Formula: see text] (the annihilation graph of the commutator poset [lattice] [Formula: see text] with respect to an element [Formula: see text]) is a complete bipartite graph together with some of its other graph-theoretic properties. In contrast to the case of rings, we construct a commutator poset whose [Formula: see text] contains a cut-point. We provide some examples to show that some conditions are not superfluous assumptions. We also give some examples of a large class of lattices, such as the lattice of ideals of a commutative ring, the lattice of normal subgroups of a group, and the lattice of all congruences on an algebra in a variety (congruence modular variety) by using the commutators as the multiplicative binary operation on these lattices. This shows that how the commutator theory can define and unify many zero-divisor type graphs of different algebraic structures as a special case of this paper.

An application of subgroup lattices

We give a lattice theoretic proof of the well-known result that a finite group G is cyclic iff G has at most one subgroup of each order dividing |G|. Consequently, we show that a division ring D is a field iff D has at most one maximal subfield.

Hyperbolic crystallography of two-periodic surfaces and associated structures.

This paper describes the families of the simplest, two-periodic constant mean curvature surfaces, the genus-two HCB and SQL surfaces, and their isometries. All the discrete groups that contain the translations of the genus-two surfaces embedded in Euclidean three-space modulo the translation lattice are derived and enumerated. Using this information, the subgroup lattice graphs are constructed, which contain all of the group-subgroup relations of the aforementioned quotient groups. The resulting groups represent the two-dimensional representations of subperiodic layer groups with square and hexagonal supergroups, allowing exhaustive enumeration of tilings and associated patterns on these surfaces. Two examples are given: a two-periodic [3,7]-tiling with hyperbolic orbifold symbol {\sf {2223}} and a {\sf {22222}} surface decoration.