Lattice Of Normal Subgroups Research Articles

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.

Lattice characterization of some classes of groups by series of subgroups

In this paper, we characterize several classes of groups by the properties of their weak congruence lattices. Namely, we give necessary and sufficient conditions for the weak congruence lattice of a group, under which this group is a [Formula: see text]-group, [Formula: see text]-group, metacyclic, cocyclic, hyperabelian, polycyclic, [Formula: see text]-group and [Formula: see text]-group. We also discuss groups for which all subgroups are simple, like the Tarski monsters, and we show that they represent a class of non-Dedekind groups with a particular embedding property for lattices of normal subgroups. All the mentioned characterized groups are related to (different) series of subgroups, and we represent these series as chains of intervals in the weak congruence lattice. The corresponding conditions are formulated in a purely lattice-theoretic language.

Normal subgroups in the group of column-finite infinite matrices

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

A note on the normal subgroup lattice of ultraproducts of finite quasisimple groups

Abel Stolz and Andreas Thom [Proc. Lond. Math. Soc. 108 (2014), pp. 73–102] stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement in this case and point out a version of \enquote*relative bounded normal generation for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of such groups.

Construction of 2-Gyrogroups in Which Every Proper Subgyrogroup Is Either a Cyclic or a Dihedral Group

In this paper, a 2-gyrogroup G(n) of order 2n, n≥3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup lattice of G(n) are isomorphic to the subgroup lattice and normal subgroup lattice of the dihedral group of order 2n, which causes us to use the name dihedral gyrogroup for this class of gyrogroups of order 2n. Moreover, all proper subgyrogroups of G(n) are subgroups.

On Normal Subgroups Lattice of Dihedral Group

In this paper, we obtain subgroup and normal subgroup lattices of dihedral group . we also give the Hasse diagrams of these lattices of D2p .

Groups whose lattices of normal subgroups are factorial

We prove that the groups G for which the lattice of normal subgroups N(G) is factorial are exactly the UND-groups, that is the groups for which every normal subgroup have a unique normal complement, with finite length.

Classifying and counting fuzzy normal subgroups by a new equivalence relation

The main goal of this paper is to give an explicit formula for the number of fuzzy normal subgroups of some finite non-abelian groups by using the new equivalence relation presented in [Fuzzy Sets and Systems, 289 (2016) 113-121]. Precisely, for a group G this new equivalence is induced by an action of the automorphism group Aut(G) associated with G on the lattice of fuzzy normal subgroups of G. In fact, this new equivalence relation generalizes the natural equivalence relation defined on the lattice of fuzzy normal subgroups.

Schurity and separability of quasiregular coherent configurations

A permutation group is said to be quasiregular if each of its transitive constituents is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with the homogeneous components have distributive lattices of normal subgroups.

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.

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.

A New Class of Generalized Thompson's Groups and Their Normal Subgroups

For a given group G and a homomorphism ϕ: G → G × G, we construct groups ℱϕ(G), 𝒯ϕ(G), and 𝒱ϕ(G) that blend Thompson's groups F, T, and V with G, respectively. Furthermore, we describe the lattice of normal subgroups of the groups ℱΔ(G), where Δ: G → G × G is the diagonal homomorphism, Δ(g) = (g, g).

Transposable character tables and group duality

AbstractOne way of expressing the self-duality $A\cong {\rm Hom}(A,\mathbb{C})$ of Abelian groups is that their character tables are self-transpose (in a suitable ordering). In this paper we extend the duality to some noncommutative groups considering when the character table of a finite group is close to being the transpose of the character table for some other group. We find that groups dual to each other have dual normal subgroup lattices. We show that our concept of duality cannot work for non-nilpotent groups and we describe p-group examples.

Imprimitivity systems and lattices of normal subgroups in D-hyperoctahedral groups

We study D-hyperoctahedral groups, diagonal inductive limits of hyperoctahedral groups. Also, we describe the Z 2-modules of periodic sequences over diagonal limits of symmetric groups under their action on the elements of a Z 2-module by permutation of coordinates. The imprimitivity systems for D-hyperoctahedral groups are characterized, and a full description of the lattices of their normal subgroups is given.

On the lattice of normal subgroups in ultraproducts of compact simple groups

We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion.

Characterization of p-valenced schemes of order [formula omitted

In [P.-H. Zieschang, Association schemes in which the thin residue is a finite cyclic group, J. Algebra 324 (2010) 3572–3578], it was shown that association schemes are schurian if their thin residues are finite groups and have a distributive normal subgroup lattice. However, in the case that their thin residues are elementary abelian groups, they do not have a distributive normal subgroup lattice. Actually, they are not necessarily schurian due to the non-schurian quasi-thin scheme of order 28. In this paper, we investigate association schemes such that their thin residues are elementary abelian groups of order p2. It is shown that these schemes are schurian if the number of elements with valency p is 2p and the factor scheme over the thin residue is a cyclic group of prime order. We also construct non-schurian p-schemes of order p3.

Solitary quotients of finite groups

Abstract We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.

Conditions under which a lattice is isomorphic to the subgroup lattice of an abelian group

In this paper, we provide necessary and sufficient conditions under which a lattice is isomorphic to the subgroup lattice of an arbitrary abelian group. We also give necessary and sufficient conditions for a lattice L, to be isomorphic to the normal subgroup lattice of an arbitrary group.

Pseudocomplementation in (Normal) Subgroup Lattices

The goal of this article is to study finite groups admitting a pseudocomplemented subgroup lattice (PK-groups) or a pseudocomplemented normal subgroup lattice (PKN-groups). In particular, we obtain a complete classification of finite PK-groups and of finite nilpotent PKN-groups. We also study groups with a Stone normal subgroup lattice, and we classify finite groups for which every subgroup has a Stone normal subgroup lattice. Finally, we obtain a complete classification of finite groups for which every subgroup is monolithic.

Association schemes in which the thin residue is a finite cyclic group

In this article, we investigate association schemes S (on finite sets) in which each element s satisfies s s ⁎ s = { s } . It is shown that these schemes are schurian if the partially ordered set of the intersections of the closed subsets s ⁎ s of S with s ∈ S is distributive. (A scheme is said to be schurian if it arises (in a well-understood way) from a transitive permutation group.) It is also shown that, if these schemes are schurian, the transitive permutation group from which they arise have subnormal one-point stabilizers. As a consequence of the first result one obtains that schemes are schurian if their thin residue is thin and has a distributive normal closed subset lattice (normal subgroup lattice). This implies, for instance, that schemes are schurian if their thin residue is a cyclic group.