Lattice Of Subgroups Research Articles

Subnormality and the Chermak–Delgado lattice

The Chermak–Delgado lattice of a finite group [Formula: see text] is a sublattice of the subgroup lattice of [Formula: see text] that has attracted interest since its discovery. In this paper, we show that every subgroup of [Formula: see text] in the Chermak–Delgado lattice is subnormal in [Formula: see text] with subnormal depth bounded by both the depth and height function of the Chermak–Delgado lattice; we provide a nontrivial example showing that our bounds are sharp. We also show that determining whether a given subgroup [Formula: see text] is in the Chermak–Delgado lattice can be decided by examining only those subgroups of [Formula: see text] that are comparable with [Formula: see text].

Classifying uniformly generated groups

A finite group G is called uniformly generated, if whenever there is a (strictly ascending) chain of subgroups then d is the minimal number of generators of G. Our main result classifies the uniformly generated groups without using the simple group classification. These groups are related to finite projective geometries by a result of Iwasawa on subgroup lattices.

On two sublattices of the subgroup lattice of a finite group

Abstract Let 𝔉 {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let ℒ ⁢ ( G ) {\mathcal{L}(G)} be the lattice of all subgroups of G. A chief H / K {H/K} factor of G is 𝔉 {\mathfrak{F}} -central in G if ( H / K ) ⋊ ( G / C G ⁢ ( H / K ) ) ∈ 𝔉 {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . Let ℒ c ⁢ 𝔉 ⁢ ( G ) {\mathcal{L}_{c\mathfrak{F}}(G)} be the set of all subgroups A of G such that every chief factor H / K {H/K} of G between A G {A_{G}} and A G {A^{G}} is 𝔉 {\mathfrak{F}} -central in G; ℒ 𝔉 ⁢ ( G ) {\mathcal{L}_{\mathfrak{F}}(G)} denotes the set of all subgroups A of G with A G / A G ∈ 𝔉 {A^{G}/A_{G}\in\mathfrak{F}} . We prove that the set ℒ c ⁢ 𝔉 ⁢ ( G ) {\mathcal{L}_{c\mathfrak{F}}(G)} and, in the case when 𝔉 {\mathfrak{F}} is a Fitting formation, the set ℒ 𝔉 ⁢ ( G ) {\mathcal{L}_{\mathfrak{F}}(G)} are sublattices of the lattice ℒ ⁢ ( G ) {\mathcal{L}(G)} . We also study conditions under which the lattice ℒ c ⁢ 𝔑 ⁢ ( G ) {\mathcal{L}_{c\mathfrak{N}}(G)} and the lattice of all subnormal subgroup of G are modular.

Galois actions on analytifications and tropicalizations

This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of 1-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin and Zharkov. We also prove two results which say that under minimal hypotheses on a variety X 0 over a Henselian field K 0 , Galois-equivariant tropicalizations carry all of the arithmetic structure of X 0 . Namely, (1) the Galois-orbit of any point of X 0 valued in the separable closure of K 0 is reproduced faithfully as a Galois-set inside some Galois-equivariant tropicalization of our variety. (2) The Berkovich analytification of X 0 over the separable closure of K 0 , equipped with its canonical Galois-action, is the inverse limit of all Galois-equivariant tropicalizations of our variety.

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.

Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum

We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.

Characteristic subgroup lattices and Hopf–Galois structures

The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].

Fundamental groups of locally connected subsets of the plane

We show that every homomorphism from the fundamental group of a one-dimensional Peano continuum to the fundamental group of a planar Peano continuum is induced by a continuous map up to conjugation. The set of points at which a planar Peano continuum X is not locally simply connected, which we denote by B(X), is determined solely by the algebraic structure of its fundamental group. Furthermore, we demonstrate how to reconstruct the topological structure of B(X) using only the subgroup lattice of its fundamental group.

Finite groups generated in low real codimension

We study the intersection lattice of the arrangement AG of subspaces fixed by subgroups of a finite linear group G. When G is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of G. We generalize the notion of finite reflection groups. We say that a group G is generated (resp. strictly generated) in codimension k if it is generated by its elements that fix point-wise a subspace of codimension at most k (resp. precisely k).We prove that the alternating subgroup Alt(W) of a reflection group W is strictly generated in codimension two. Further, we compute the intersection lattice of all finite subgroups of GL3(R), and moreover, we emphasize the groups that are “minimally generated in real codimension two”, i.e., groups that are strictly generated in codimension two but have no real reflection representations. We also provide several examples of groups generated in higher codimension.

The Möbius function of PSU(3, 2^{2^n})

Let G be the simple group PSU(3, 2 2 n ) , n > 0 . For any subgroup H of G , we compute the Mobius function μ L ( H , G ) of H in the subgroup lattice L of G , and the Mobius function μ L ([ H ], [ G ]) of [ H ] in the poset L of conjugacy classes of subgroups of G . For any prime p , we provide the Euler characteristic of the order complex of the poset of non-trivial p -subgroups of G .

Imaginary cone and reflection subgroups of Coxeter groups

The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice is explicitly described in terms of the lattice of facial reflection subgroups and it is shown that the Tits cone and imaginary cone are related by a duality closely analogous to the standard duality for polyhedral cones, even though neither of them is a closed cone in general. Some of these results have application, to be given in sequels to this paper, to dominance order of Coxeter groups, associated automata, and construction of modules for generic Iwahori-Hecke algebras.

Entropy Inequalities for Lattices.

We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice.

Test map characterizations of local properties of fundamental groups

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any path-connected metric space [Formula: see text], and a subgroup [Formula: see text], we characterize the unique path lifting property relative to [Formula: see text] in terms of a new closure operator on the [Formula: see text]-subgroup lattice that is induced by maps from a fixed “test” domain into [Formula: see text]. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS

A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

Convex $L$-lattice subgroups in $L$-ordered groups

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result about the set of all convex subgroups of a lattice ordered group and prove that $C(G)$, the set of all convex $L$-lattice subgroups of an $L$-ordered group $G$, is an $L$-complete lattice on height one. Then we use these objects to construct the quotient $L$-ordered groups and state some related results.

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.

Large characteristic subgroups with modular subgroup lattice

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.

A New Metatheorem and Subdirect Product Theorem for L-Subgroups

ABSTRACTThis paper is a continuation of the work of Tom Head ‘Metatheorem for deriving fuzzy theorems from crisp versions’. The concept of natural extension is introduced which is then applied in the development of a new metatheorem in L-setting, where L is a complete chain. The application of this theorem is demonstrated through the notions of generated L-subgroups and commutator L-subgroups. Moreover, a new subdirect product theorem is developed wherein it is demonstrated that for a group G, the L-subgroup lattice can be represented as a subdirect product of copies of its associated lattice of crisp subgroups. The significance of this theorem is exhibited by applying it to a characterization of generalized cyclic groups in terms of the distributivity of the lattice of L-subgroups of a group.

Symmetric products and subgroup lattices

Let G be a finite group. We show that the rational homotopy groups of symmetric products of the G-equivariant sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of G.

On Boolean intervals of finite groups

We prove a dual version of Øystein Ore's theorem on distributive intervals in the subgroup lattice of finite groups, having a nonzero dual Euler totient φˆ. For any Boolean group-complemented interval, we observe that φˆ=φ≠0 by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that φˆ is always nonzero for Boolean intervals. In order to investigate it, we prove that for any Boolean group-complemented interval [H,G], the graded coset poset Pˆ=Cˆ(H,G) is Cohen–Macaulay and the nontrivial reduced Betti number of the order complex Δ(P) is φˆ, so nonzero. We deduce that these results are true beyond the group-complemented case with |G:H|<32. One observes that they are also true when H is a Borel subgroup of G.