Lattice Of Subgroups Research Articles

Some density results involving the average order of a finite group

Let be the average of the element orders of a finite group G. A research topic concerning this quantity is understanding the relation between and , where H is a subgroup of G. Let be the class of finite nilpotent groups and let be the subgroup lattice of G. In this paper, we show that the set is dense in . Other density results are outlined throughout the paper.

Tensor induction for M-Burnside rings

Let G be a finite group and H a subgroup. The M-Burnside ring ΩM(H) arising from a monoid functor M for G is a generalization of the Burnside ring Ω(H). We present a certain hypothesis on M such that if M satisfies the hypothesis, then there is a multiplicative map θHG:ΩM(H)→ΩM(G) derived from the tensor induction jndHG:Ω(H)→Ω(G). Let A be a finite abelian G-group. For the monoid functor HA assigning each K≤G the 1st cohomology group H1(K,A) of K with coefficients in A, the HA-Burnside ring ΩHA(H) is isomorphic to the ring of monomial representations of H with coefficients in A. We show that HA satisfies the hypothesis and the associated multiplicative map defines the known tensor (or multiplicative) induction for the rings of monomial representations. Let S be a finite commutative G-monoid. The monoid functor CS assigning each K≤G the set of K-invariants in S satisfies the hypothesis. To extend this fact, we introduce the concept of hereditary monoid functors which satisfy the hypothesis. The monoid functor MS∘ assigning each K≤G the set of normal subgroups of K is a hereditary monoid functor for G on the subgroup lattice of G. In short, we present tensor inductions for HA-, CS-, and MS∘-Burnside rings derived from the tensor induction for Burnside rings.

On the Uniqueness of Lattice Characterization of Groups

We analyze the problem of the uniqueness of characterization of groups by their weak congruence lattices. We discuss the possibility that the same algebraic lattice L acts as a weak congruence lattice of a group in more than one way, so that the corresponding diagonals are represented by different elements of L. If this is impossible, that is, if L can be interpreted as a weak congruence lattice of a group in a single way, we say that L is a sharp lattice. We prove that groups in many classes have a sharp weak congruence lattice. In particular, we analyze connections among isomorphisms of subgroup lattices of groups and isomorphisms of their weak congruence lattices. Summing up, we prove that there is a one-to-one correspondence between many known classes of groups and lattice-theoretic properties associated with each of these classes. Finally, an open problem is formulated related to the uniqueness of the element corresponding to the diagonal in the lattice of weak congruences of a group.

The A-Möbius function of a finite group

The Möbius function of the subgroup lattice of a finite group G has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let A be a subgroup of the automorphism group Aut(G) of a finite group G and denote by CA(G) the set of A-conjugacy classes of subgroups of G. For H ≤ G let [H]A = { Ha ∣ a∈ A} be the element of CA(G) containing H. We may define an ordering in CA(G) in the following way: [H]A ≤ [K]A if Ha ≤ K for some a ∈ A. We consider the Möbius function μA of the corresponding poset and analyse its properties and possible applications.

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.

On some subgroup lattices of dihedral, alternating and symmetric groups

On some subgroup lattices of dihedral, alternating and symmetric groups

On subpolygroup commutativity degree of finite polygroups

<abstract><p>Probabilistic group theory is concerned with the probability of group elements or group subgroups satisfying certain conditions. On the other hand, a polygroup is a generalization of a group and a special case of a hypergroup. This paper generalizes probabilistic group theory to probabilistic polygroup theory. In this regard, we extend the concept of the subgroup commutativity degree of a finite group to the subpolygroup commutativity degree of a finite polygroup $ P $. The latter measures the probability of two random subpolygroups $ H, K $ of $ P $ commuting (i.e., $ HK = KH $). First, using the subgroup commutativity table and the subpolygroup commutativity table, we present some results related to the new defined concept for groups and for polygroups. We then consider the special case of a polygroup associated to a group. We study the subpolygroup lattice and relate this to the subgroup lattice of the base group; this includes deriving an explicit formula for the subpolygroup commutativity degree in terms of the subgroup commutativity degree. Finally, we illustrate our results via non-trivial examples by applying the formulas that we prove to the associated polygroups of some well-known groups such as the dihedral group and the symmetric group.</p></abstract>

Normal structure of isotropic reductive groups over rings

The paper studies the lattice of subgroups of an isotropic reductive group G(R) over a commutative ring R, normalized by the elementary subgroup E(R). We prove the sandwich classification theorem for this lattice under the assumptions that the isotropic rank of G is at least 2 and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E(R,q)⩽…⩽C(R,q) parametrized by the ideals q of R, where E(R,q) denotes the relative elementary subgroup and C(R,q) is the inverse image of the center under the natural homomorphism G(R)→G(R/q). The main ingredients of the proof are the “level computation” by the first author and the generic element method developed by the second author.

FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory $${\varPhi }$$ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra $$H^{\text {cut}}_{\text {FG}}({\varPhi })$$ and lattices of intermediate structures. Feynman diagrams in $$H_{\text {FG}}({\varPhi })$$ are applied to describe states in $${\varPhi }$$ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space $$\mathcal {S}^{{\varPhi },M \subseteq [0,\infty )}_{\text {graphon}}([0,1])$$ . We associate Hopf subalgebras of $$H_{\text {FG}}({\varPhi })$$ generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green’s functions of $${\varPhi }$$ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales.

How robustly can you predict the future?

Abstract Hardin and Taylor proved that any function on the reals—even a nowhere continuous one—can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman, who provided upper and lower frontiers (in the subgroup lattice of $\mathrm{Homeo}^+(\mathbb {R})$ ) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder’s Theorem (that every Archimedean group is abelian).

Aggregation of T-subgroups

We study when an aggregation function acting on an n-tuple of T-subgroups preserves the structure of T-subgroup. First we need to consider that there are two known definitions applicable to the aggregation of structures on fuzzy sets. These two notions differ in the domain of the aggregated structure. It is known that for indistinguishability operators, pseudometrics, quasi-pseudometrics among others, the aggregation functions that preserve these structures are the same with both definitions. However this is not the case for quasi-metrics. In this line we study the aggregation of T-subgroups with both definitions and their implications. We see that aggregation functions may preserve the structure of T-subgroups with one definition but not with the other. However, under adequate restrictions, the aggregation functions preserving the structure of T-subgroups are the same with either definition. We also show that the results depend on the structure of the subgroup lattice of the ambient group, the particular T-subgroups being aggregated, or the aggregation function.

FINITE GROUPS WITH LARGE CHERMAK–DELGADO LATTICES

AbstractGiven a finite group G, we denote by $L(G)$ the subgroup lattice of G and by ${\cal CD}(G)$ the Chermak–Delgado lattice of G. In this note, we determine the finite groups G such that $|{\cal CD}(G)|=|L(G)|-k$ , for $k=1,2$ .

Projective classes as images of accessible functors

Abstract We are dealing with projective classes (in short $\textrm {PC}$) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {RPC}(\mathbin {\mathscr {L}}_{\infty \lambda })$ for any infinite cardinal $\lambda $, and that if $\lambda $ is singular, then $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda ^+})$. If $\lambda $ is regular, then a class of structures over a $\lambda $-ary vocabulary is $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })$-definable iff it is the image of a $\lambda $-continuous functor on a $\lambda $-accessible category; we also provide separating counterexamples for the non $\lambda $-ary case. We prove that many $\textrm {PC}$ classes of structures, previously known not to be closed under elementary equivalence over any $\mathbin {\mathscr {L}}_{\infty \lambda }$, are not even $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. Those classes arise from diverse contexts including convex $\ell $-subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K$_0$-theory of rings, coordinatization of sectionally complemented modular lattices and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is $\textrm {PC}$ but not $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. We also provide a negative solution to a problem, raised in 2011 by Gillibert and the author, asking whether essential surjectivity of a ‘well-behaved’ functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets.

Subgroups Inclusions in 3-Factors Direct Product

The aim of this paper is to use a correspondent theorem to characterize containment of a degenerate $2$-factor injective subdirect products. Namely, let $\Omega,\Lambda$ be degenerate 2-factor injective subdirect products of $ M_{1}\times M_{2}\times M_{3}$, we provide necessary and sufficient conditions for $\Omega\leq \Lambda.$ Based on a decomposition of the inclusion order on the subgroup lattice of a subdirect product as a relation product of three smaller partial orders, we induce a matrix product of three incidence matrices.

Groups whose Chermak–Delgado lattice is a subgroup lattice of an abelian group

AbstractThe Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we prove that, for any finite abelian group A, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of A.

C1 actions on manifolds by lattices in Lie groups

In this paper we study Zimmer's conjecture for $C^{1}$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in ${\rm SL}(n, {{\mathbb {R}}})$, the dimensional bound is sharp.

A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds

Let \(M= \Gamma \setminus \mathbb {H}_d\) be a compact quotient of the d-dimensional Heisenberg group \(\mathbb {H}_d\) by a lattice subgroup \(\Gamma \). We show that the eigenvalue counting function \(N^\alpha \left( \lambda \right) \) for any fixed element of a family of second order differential operators \(\left\{ \mathcal {L}_\alpha \right\} \) on M has asymptotic behavior \(N^\alpha \left( \lambda \right) \sim C_{d,\alpha } {\text {vol}}\left( M\right) \lambda ^{d + 1}\), where \(C_{d,\alpha }\) is a constant that only depends on the dimension d and the parameter \(\alpha \). As a consequence, we obtain an analog of Weyl’s law (both on functions and forms) for the Kohn Laplacian on M. Our main tools are Folland’s description of the spectrum of \({\mathcal {L}}_{\alpha }\) and Karamata’s Tauberian theorem.

Groups whose Chermak–Delgado lattice is a subgroup lattice of an elementary abelian p-group

The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we focus on finite groups whose Chermak–Delgado lattice is a subgroup lattice of an elementary abelian p-group. We prove that such groups are nilpotent of class 2. We also prove that, for any elementary abelian p-group E, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of E.

How the norm one positive definite functions determine a finite group

We study a finite (discrete) group G through the information we obtain fromP1(G)={〈π(⋅)ξ,ξ〉:π:G→U(H)is unitary,ξ∈H,‖ξ‖=1} of norm one positive definite functions of G, arising from matrix coefficients of any unitary representation π. Below we summarize our results.(a)Knowing P1(G) as a set of functions, when G is a finite abelian group we can determine G≅∏jZpjrj as a direct product of its cyclic subgroups of prime power orders.(b)Knowing P1(G) as a multiplicative semigroup, we can construct the subgroup lattice L(G) of G. With L(G) in stock, we can tell if G is cyclic, simple, perfect, solvable, supersolvable, or nilpotent. When G′ is a finite simple group with P1(G′)≅P1(G) as multiplicative semigroups, we show that G′≅G as groups.(c)Knowing P1(G) as a compact convex set, we can construct the group von Neumann algebra vN(G) of G. Consequently, when G′ is another finite group with P1(G)≅P1(G′) as convex sets, we show that vN(G)≅vN(G′) as von Neumann algebras. In particular, we can tell if G is abelian.

Infinite Locally Finite Simple Groups with Many Complemented Subgroups

We prove that the following families of (infinite) groups have complemented subgroup lattice‎: ‎alternating groups‎, ‎finitary symmetric groups‎, ‎Suzuki groups over an infinite locally finite field of characteristic $2$‎, ‎Ree groups over an infinite locally finite field of characteristic~$3$‎. ‎We also show that if the Sylow primary subgroups of a locally finite simple group $G$ have complemented subgroup lattice‎, ‎then this is also the case for $G$‎.