scholarly journals Full automorphism group of commuting and non-commuting graph of dihedral and symmetric groups

2018 ◽  
Vol 1028 ◽  
pp. 012112
Author(s):  
Abdussakir ◽  
Dini Chandra Aulia Putri ◽  
Ziyadatur Rohmah Fadhillah
Keyword(s):  
Automorphism Group ◽  
Symmetric Groups ◽  
Full Automorphism Group ◽  
Commuting Graph

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 On diameters estimations of the commuting graphs of Sylow $p$-subgroups of the symmetric groups

2013 ◽  
Vol 5 (1) ◽  
pp. 70-78 ◽  
Author(s):  
Yu.Yu. Leshchenko ◽  
L.V. Zoria
Keyword(s):  
Undirected Graph ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Central Elements

The commuting graph of a group $G$ is an undirected graph whose vertices are non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if and only if $xy=yx$. This article deals with the properties of the commuting graphs of Sylow $p$-subgroups of the symmetric groups. We define conditions of connectedness of respective graphs and give estimations of the diameters if graph is connected.

scholarly journals A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

2019 ◽  
pp. 729
Author(s):  
Mahsa Mirzargar
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
The Other ◽  
Power Graph ◽  
Commuting Graph ◽  
Vertex Set ◽  
Delta G ◽  
Nite Group

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

scholarly journals Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1178
Author(s):  
Athirah Nawawi ◽  
Sharifah Kartini Said Husain ◽  
Muhammad Rezal Kamel Ariffin
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Commutativity Property

A commuting graph is a graph denoted by C ( G , X ) where G is any group and X, a subset of a group G, is a set of vertices for C ( G , X ) . Two distinct vertices, x , y ∈ X , will be connected by an edge if the commutativity property is satisfied or x y = y x . This study presents results for the connectivity of C ( G , X ) when G is a symmetric group of degree n, Sym ( n ) , and X is a conjugacy class of elements of order three in G.

scholarly journals Cohomological finiteness conditions and centralisers in generalisations of Thompson’s group V

2016 ◽  
Vol 28 (5) ◽  
pp. 909-921 ◽  
Author(s):  
Conchita Martínez-Pérez ◽  
Francesco Matucci ◽  
Brita E. A. Nucinkis
Keyword(s):  
Automorphism Group ◽  
Finiteness Conditions ◽  
Group V ◽  
Full Automorphism Group ◽  
Finite Subgroups ◽  
Thompson’S Group

AbstractWe consider generalisations of Thompson’s group V, denoted by ${V_{r}(\Sigma)}$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, ${V_{r}(\Sigma)}$ is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type ${\operatorname{F}_{\infty}}$ and that this implies that also centralisers of finite subgroups are of type ${\operatorname{F}_{\infty}}$.

scholarly journals Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer

2019 ◽  
Vol 17 (1) ◽  
pp. 513-518
Author(s):  
Hailin Liu
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Frobenius Groups ◽  
Full Automorphism Group

Abstract A Cayley graph Γ is said to be arc-transitive if its full automorphism group AutΓ is transitive on the arc set of Γ. In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of Frobenius groups with soluble vertex stabilizer.

Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

2020 ◽  
pp. 2150214
Author(s):  
Xinlei Wang ◽  
Dein Wong ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Zero Divisor ◽  
Symmetric Groups ◽  
Directed Edge ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisor Graph ◽  
Vertex Set ◽  
The One ◽  
Definition Of

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

scholarly journals Rooted trees, strong cofinality and ample generics

2012 ◽  
Vol 154 (2) ◽  
pp. 213-223
Author(s):  
MACIEJ MALICKI
Keyword(s):  
Automorphism Group ◽  
Open Subgroup ◽  
Rooted Trees ◽  
Full Automorphism Group

AbstractWe characterize those countable rooted trees with non-trivial components whose full automorphism group has uncountable strong cofinality, and those whose full automorphism group contains an open subgroup with ample generics.

A 2-ARC TRANSITIVE PENTAVALENT CAYLEY GRAPH OF 

2016 ◽  
Vol 93 (3) ◽  
pp. 441-446 ◽  
Author(s):  
BO LING ◽  
BEN GONG LOU
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Discrete Math ◽  
Simple Groups ◽  
Alternating Group ◽  
Full Automorphism Group ◽  
Symmetric Graphs

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

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