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.

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 groups and Picard groups of additive full subcategories

2010 ◽  
Vol 107 (1) ◽  
pp. 5 ◽  
Author(s):  
Naoya Hiramatsu ◽  
Yuji Yoshino
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Full Subcategory ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Picard Group ◽  
Picard Groups

We study category equivalences between additive full subcategories of module categories over commutative rings. And we are able to define the Picard group of additive full subcategories. The aim of this paper is to study the properties of the Picard groups and show that the automorphism group of an additive full subcategory is a semi-direct product of the Picard group with the group of algebra automorphisms of the ring.

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.

scholarly journals Semidirect product groups with Abelian automorphism groups

1987 ◽  
Vol 42 (1) ◽  
pp. 84-91 ◽  
Author(s):  
M. J. Curran
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Semidirect Product ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Infinite Family ◽  
Central Automorphism ◽  
Nonabelian Group ◽  
Semidirect Product Groups

AbstractMiller's group of order 64 is a smallest example of a nonabelian group with an abelian automorphism group, and is the first in an infinite family of such groups formed by taking the semidirect product of a cyclic group of order 2m (m ≥ 3) with a dihedral group of order 8. This paper gives a method for constructing further examples of non abelian 2-groups which have abelian automorphism groups. Such a 2-group is the semidirect product of a cyclic group and a special 2-group (satisfying certain conditions). The automorphism group of this semidirect product is shown to be isomorphic to the central automorphism group of the corresponding direct product. The conditions satisfied by the special 2-group are determined by establishing when this direct product has an abelian central automorphism group.

Algebras with Transitive Automorphism Groups

1986 ◽  
Vol 29 (2) ◽  
pp. 224-226
Author(s):  
L. G. Sweet ◽  
J. A. MacDougall
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Automorphism Groups ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Transitive Automorphism ◽  
Cyclic Groups ◽  
Finite Dimensional

AbstractLet A be a finite dimensional algebra (not necessarily associative) over a field, whose automorphism group acts transitively. It is shown that K = GF(2) and A is a Kostrikin algebra. The automorphism group is determined to be a semi-direct product of two cyclic groups. The number of such algebras is also calculated.

CONJUGATION IN TREE AUTOMORPHISM GROUPS

2001 ◽  
Vol 11 (05) ◽  
pp. 529-547 ◽  
Author(s):  
PIOTR W. GAWRON ◽  
VOLODYMYR V. NEKRASHEVYCH ◽  
VITALY I. SUSHCHANSKY
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Rooted Tree ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Rooted Trees ◽  
Full Description ◽  
Finite Tree ◽  
Locally Finite ◽  
Infinite Sequences

It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

scholarly journals Automorphism Groups of a Class of Cubic Cayley Graphs on Symmetric Groups

2017 ◽  
Vol 24 (04) ◽  
pp. 541-550
Author(s):  
Xueyi Huang ◽  
Qiongxiang Huang ◽  
Lu Lu
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Full Automorphism Group ◽  
Normal Cayley Graph ◽  
The Right

Let Sndenote the symmetric group of degree n with n ≥ 3, S = { cn= (1 2 ⋯ n), [Formula: see text], (1 2)} and Γn= Cay(Sn, S) be the Cayley graph on Snwith respect to S. In this paper, we show that Γn(n ≥ 13) is a normal Cayley graph, and that the full automorphism group of Γnis equal to Aut(Γn) = R(Sn) ⋊ 〈Inn(ϕ) ≅ Sn× ℤ2, where R(Sn) is the right regular representation of Sn, ϕ = (1 2)(3 n)(4 n−1)(5 n−2) ⋯ (∊ Sn), and Inn(ϕ) is the inner isomorphism of Sninduced by ϕ.

scholarly journals CI-Property for Decomposable Schur Rings over an Abelian Group

2019 ◽  
Vol 26 (01) ◽  
pp. 147-160 ◽  
Author(s):  
István Kovács ◽  
Grigory Ryabov
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Wreath Product ◽  
Direct Product ◽  
Elementary Abelian Group ◽  
Sufficient Condition ◽  
Schur Ring ◽  
Generalized Wreath Product ◽  
A Finite Group ◽  
Ci Property

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a CI-Schur ring. By using this condition we offer short proofs for some known results on the CI-property for decomposable Schur rings over an elementary abelian group of rank at most 5.

scholarly journals The rank of the semigroup of transformations stabilising a partition of a finite set

2015 ◽  
Vol 159 (2) ◽  
pp. 339-353 ◽  
Author(s):  
JOÃO ARAÚJO ◽  
WOLFRAM BENTZ ◽  
JAMES D. MITCHELL ◽  
CSABA SCHNEIDER
Keyword(s):  
Representation Theory ◽  
Upper Bound ◽  
Symmetric Groups ◽  
Complex Problem ◽  
Minimum Size ◽  
Generating Sets ◽  
Arbitrary Partition ◽  
Finite Set ◽  
Minimum Number ◽  
Semigroup Of Transformations

AbstractLet $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$.In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture.The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.The paper ends with a number of problems for experts in group and semigroup theories.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

Sign in / Sign up

Export Citation Format

Share Document