scholarly journals Frankl's Conjecture for Subgroup Lattices

10.37236/6248 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Alireza Abdollahi ◽  
Russ Woodroofe ◽  
Gjergji Zaimi
Keyword(s):  
Finite Group ◽  
Subgroup Lattice ◽  
Independent Interest ◽  
Technical Result ◽  
Semimodular Lattices ◽  
Subgroup Lattices ◽  
Frankl’S Conjecture

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.

On two sublattices of the subgroup lattice of a finite group

2019 ◽  
Vol 22 (6) ◽  
pp. 1035-1047 ◽  
Author(s):  
Zhang Chi ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Subnormal Subgroup ◽  
Subgroup Lattice ◽  
Chief Factor ◽  
A Finite Group ◽  
Fitting Formation

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

scholarly journals A connection between the number of subgroups and the order of a finite group

2020 ◽  
pp. 2250001
Author(s):  
Mihai-Silviu Lazorec
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Subgroup Lattice ◽  
Minimum Value ◽  
A Finite Group

For a finite group [Formula: see text], we associate the quantity [Formula: see text], where [Formula: see text] is the subgroup lattice of [Formula: see text]. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of [Formula: see text] on the class of [Formula: see text]-groups of order [Formula: see text], where [Formula: see text] is an integer. We show that the set containing the quantities [Formula: see text], where [Formula: see text] is a finite (abelian) group, is dense in [Formula: see text] Finally, we consider [Formula: see text] to be a function on [Formula: see text] and we indicate some of its properties, the main result being the classification of finite abelian [Formula: see text]-groups [Formula: see text] satisfying [Formula: see text]

scholarly journals THE TOM DIECK SPLITTING THEOREM IN EQUIVARIANT MOTIVIC HOMOTOPY THEORY

2021 ◽  
pp. 1-70
Author(s):  
DAVID GEPNER ◽  
JEREMIAH HELLER
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Homotopy Theory ◽  
Independent Interest ◽  
Splitting Theorem ◽  
Motivic Homotopy Theory ◽  
Group A ◽  
Motivic Homotopy ◽  
A Finite Group

Abstract We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

On the Diameters of Commuting Graphs Arising from Random Skew-Symmetric Matrices

2014 ◽  
Vol 23 (3) ◽  
pp. 449-459 ◽  
Author(s):  
PETER HEGARTY ◽  
DMITRII ZHELEZOV
Keyword(s):  
Finite Group ◽  
Random Graph ◽  
Large Family ◽  
Parameter Family ◽  
Independent Interest ◽  
Symmetric Matrices ◽  
Commuting Graph ◽  
A Finite Group ◽  
Random Groups ◽  
Two Parameter

We present a two-parameter family $(G_{m,k})_{m, k \in \mathbb{N}_{\geq 2}}$, of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Rényi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

scholarly journals Seminormal and subnormal subgroup lattices for transitive permutation groups

2006 ◽  
Vol 80 (1) ◽  
pp. 45-64
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Subnormal Subgroup ◽  
Subgroup Lattice ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Transitive Groups ◽  
Lattices Of Subgroups ◽  
Subgroup Lattices ◽  
A Chain ◽  
Seminormal Subgroup ◽  
Composition Factors

AbstractVarious lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.

scholarly journals Large violations in Kochen Specker contextuality and their applications

2021 ◽  
Author(s):  
Ravishankar Ramanathan ◽  
Yuan Liu ◽  
Pawel Horodecki
Keyword(s):  
Quantum Mechanics ◽  
Present State ◽  
Hilbert Spaces ◽  
Independent Interest ◽  
Entangled State ◽  
Maximally Entangled State ◽  
Technical Result ◽  
Orthogonal Representation ◽  
State Dependent ◽  
Minimum Dimension

Abstract It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen-Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension $d \geq 2^{17}$ with the ratio of quantum value to classical bias being $O(\sqrt{d}/\log d)$. We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen-Specker proofs always consist of substructures of state-dependent contextuality proofs called $01$-gadgets or bugs. We show a one-to-one connection between $01$-gadgets in $\mathbb{C}^d$ and Hardy paradoxes for the maximally entangled state in $\mathbb{C}^d \otimes \mathbb{C}^d$. We use this connection to construct large violation $01$-gadgets between arbitrary vectors in $\mathbb{C}^d$, as well as novel Hardy paradoxes for the maximally entangled state in $\mathbb{C}^d \otimes \mathbb{C}^d$, and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in $\mathbb{R}^d$ is not a graph monotone, a result that may be of independent interest.

scholarly journals Permutability degrees of finite groups

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2165-2175
Author(s):  
D.E. Otera ◽  
F.G. Russo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subgroup Lattice ◽  
Cyclic Subgroups ◽  
A Finite Group

Given a finite group G, we introduce the permutability degree of G, as pd(G) = 1/|G| |L(G)| ?X?L(G)|PG(X)|, where L(G) is the subgroup lattice of G and PG(X) the permutizer of the subgroup X in G, that is, the subgroup generated by all cyclic subgroups of G that permute with X ? L(G). The number pd(G) allows us to find some structural restrictions on G. Successively, we investigate the relations between pd(G), the probability of commuting subgroups sd(G) of G and the probability of commuting elements d(G) of G. Proving some inequalities between pd(G), sd(G) and d(G), we correlate these notions.

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