scholarly journals Automorphism groups of some variants of lattices

2021 ◽  
Vol 13 (1) ◽  
pp. 142-148
Author(s):  
O.G. Ganyushkin ◽  
O.O. Desiateryk
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Wreath Product ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Natural Generalization ◽  
Space Lattice ◽  
Finite Vector ◽  
Finite Vector Space ◽  
Finite Set

In this paper we consider variants of the power set and the lattice of subspaces and study automorphism groups of these variants. We obtain irreducible generating sets for variants of subsets of a finite set lattice and subspaces of a finite vector space lattice. We prove that automorphism group of the variant of subsets of a finite set lattice is a wreath product of two symmetric permutation groups such as first of this groups acts on subsets. The automorphism group of the variant of the subspace of a finite vector space lattice is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the automorphism group of subspaces lattice and the second is defined by the certain set of symmetric groups.

scholarly journals Automorphism group of the variant of the lattice of partitions of a finite set

2020 ◽  
pp. 115-119
Author(s):  
O. G. Ganyushkin ◽  
O. O. Desiateryk
Keyword(s):  
Automorphism Group ◽  
Wreath Product ◽  
Direct Product ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Natural Generalization ◽  
Generating Sets ◽  
Generalized Wreath Product ◽  
Finite Set

In this paper we consider variants of the lattice of partitions of a finite set and study automorphism groups of this variants. We obtain irreducible generating sets for of the lattice of partitions of a finite set. We prove that the automorphism group of the variant of the lattice of partitions of a finite set is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the direct product of the wreaths products, such that depends on the type of the variant generating partition and the second is defined by the certain set of symmetric groups.

Orbits and Stabilizers for Solvable Linear Groups

2006 ◽  
Vol 49 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Irreducible Character ◽  
General Class ◽  
Frobenius Group ◽  
Solvable Groups ◽  
Character Degrees ◽  
Linear Groups ◽  
Finite Vector ◽  
Finite Vector Space

AbstractWe extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group G on a finite vector space V under certain conditions, to a more general class of finite solvable groups G. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.

scholarly journals On the orders of primitive linear P'-groups

1993 ◽  
Vol 48 (3) ◽  
pp. 495-521 ◽  
Author(s):  
A. Gambini Weigel ◽  
T.S. Weigel
Keyword(s):  
Vector Space ◽  
Lower Bounds ◽  
Minimal Representation ◽  
Group Order ◽  
Finite Vector ◽  
Finite Vector Space

A group G ≤ GLK(V) is called K-primitive if there exists no non-trivial decomposition of V into a sum of K-spaces which is stabilised by G. We show that if V is a finite vector space and G a K-primitive subgroup of GLK(V) whose order is coprime to |V|, we can bound the order of G by |V|log2(|V|) apart from one exception. Later we use this result to obtain some lower bounds on the number of p–singular elements in terms of the group order and the minimal representation degree.

scholarly journals On resolvability of a graph associated to a finite vector space

2019 ◽  
Vol 18 (02) ◽  
pp. 1950029
Author(s):  
U. Ali ◽  
S. A. Bokhary ◽  
K. Wahid ◽  
G. Abbas
Keyword(s):  
Vector Space ◽  
Domination Number ◽  
Exchange Property ◽  
Metric Dimension ◽  
Resolving Set ◽  
Matroid Intersection ◽  
Finite Vector ◽  
Finite Vector Space ◽  
Partition Dimension ◽  
Matroid Intersection Problem

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin. 25 (2009) 789–806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set [Formula: see text] of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with [Formula: see text] as the ground set is also solved.

scholarly journals Counting subspaces of a finite vector space — 2

Resonance ◽  
2010 ◽  
Vol 15 (12) ◽  
pp. 1074-1083 ◽  
Author(s):  
Amritanshu Prasad
Keyword(s):  
Vector Space ◽  
Finite Vector ◽  
Finite Vector Space

scholarly journals The affinity of a permutation of a finite vector space

2007 ◽  
Vol 13 (1) ◽  
pp. 80-112
Author(s):  
W. Edwin Clark ◽  
Xiang-dong Hou ◽  
Alec Mihailovs
Keyword(s):  
Vector Space ◽  
Finite Vector ◽  
Finite Vector Space

scholarly journals Covering codes of a graph associated to a finite vector space

2020 ◽  
Vol 72 (7) ◽  
pp. 952-959
Author(s):  
M. Murtaza ◽  
I. Javaid ◽  
M. Fazil
Keyword(s):  
Vector Space ◽  
Domination Number ◽  
Exchange Property ◽  
Dominating Sets ◽  
Identifying Codes ◽  
Finite Vector ◽  
Finite Vector Space ◽  
Covering Codes

UDC 512.5 In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das [Commun. Algebra, <strong>44</strong>, 3918 – 3926 (2016)], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.

Automorphism Groups of Unary Algebras on Groups

1969 ◽  
Vol 21 ◽  
pp. 1165-1171 ◽  
Author(s):  
G. H. Wenzel
Keyword(s):  
Automorphism Group ◽  
Wreath Product ◽  
Systematic Study ◽  
Automorphism Groups ◽  
Unary Algebra ◽  
Main Result Theorem ◽  
Universal Algebras ◽  
Result Theorem ◽  
The Right ◽  
Characterization Theorems

This paper presents a systematic study of the automorphism groups of those unary (universal) algebras whose carrier set G is the carrier set of some group ) and whose automorphism set contains the right translations of the latter group. These algebras appear, apart from the known classical contexts, repeatedly in characterization theorems of endomorphism semigroups (End) and automorphism groups (Aut) of algebras due to Grätzer (3; 4; 5), Makkai (7), Armbrust and Schmidt (1), Birkhoff (2), and others.Our main result (Theorem 1) constitutes an essential strengthening of a theorem of Birkhoff and represents the automorphism group of a unary algebra (where F is contained in the set of left translations of the group as wreath product of two groups that are easily determined from F and G.

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