scholarly journals Automorphism Group of the Derangement Graph

10.37236/685 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Yun-Ping Deng ◽  
Xiao-Dong Zhang
Keyword(s):  
Automorphism Group ◽  
Regular Representation ◽  
Full Automorphism Group ◽  
Sigma 1 ◽  
Inner Automorphism ◽  
The Right ◽  
Derangement Graph

In this paper, we prove that the full automorphism group of the derangement graph $\Gamma_n$ ($n\geq3$) is equal to $(R(S_n)\rtimes\hbox{Inn} (S_n))\rtimes Z_2$, where $R(S_n)$ and $\hbox{Inn} (S_n)$ are the right regular representation and the inner automorphism group of $S_n$ respectively, and $Z_2=\langle\varphi\rangle$ with the mapping $\varphi:$ $\sigma^{\varphi}=\sigma^{-1},\,\forall\,\sigma\in S_n.$ Moreover, all orbits on the edge set of $\Gamma_n$ ($n\geq3$) are determined.

Automorphisms of a Family of Cubic Graphs

2013 ◽  
Vol 20 (03) ◽  
pp. 495-506 ◽  
Author(s):  
Jin-Xin Zhou ◽  
Mohsen Ghasemi
Keyword(s):  
Automorphism Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Vertex Set ◽  
The Right ◽  
Normal Cayley Graphs

A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

scholarly journals Tetravalent Non-Normal Cayley Graphs of Order $4p$

10.37236/207 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Full Automorphism Group ◽  
The Right ◽  
Normal Cayley Graphs

A Cayley graph ${\rm Cay}(G,S)$ on a group $G$ is said to be normal if the right regular representation $R(G)$ of $G$ is normal in the full automorphism group of ${\rm Cay}(G,S)$. In this paper, all connected tetravalent non-normal Cayley graphs of order $4p$ are constructed explicitly for each prime $p$. As a result, there are fifteen sporadic and eleven infinite families of tetravalent non-normal Cayley graphs of order $4p$.

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.

Cubic graphs admitting transitive non-abelian characteristically simple groups

2011 ◽  
Vol 54 (1) ◽  
pp. 113-123 ◽  
Author(s):  
Xiao-Hui Hua ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Cayley Graph ◽  
Regular Representation ◽  
Simple Groups ◽  
Symmetric Graph ◽  
Cubic Graphs ◽  
Vertex Transitive ◽  
Transitive Subgroup ◽  
The Right

AbstractLet Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

Automorphism Groups of Connected Cubic Cayley Graphs of Order 4p

2007 ◽  
Vol 14 (02) ◽  
pp. 351-359 ◽  
Author(s):  
Chuixiang Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Three Dimensional ◽  
Group A ◽  
The Right

For a prime p, let D4p be the dihedral group 〈a,b | a2p = b2 = 1, b-1ab = a-1〉 of order 4p, and Cay (G,S) a connected cubic Cayley graph of order 4p. In this paper, it is shown that the automorphism group Aut ( Cay (G,S)) of Cay (G,S) is the semiproduct R(G) ⋊ Aut (G,S), where R(G) is the right regular representation of G and Aut (G,S) = {α ∈ Aut (G) | Sα = S}, except either G = D4p (p ≥ 3), Sβ = {b,ab,apb} for some β ∈ Aut (D4p) and [Formula: see text], or Cay (G,S) is isomorphic to the three-dimensional hypercube Q3[Formula: see text] and G = ℤ4 × ℤ2 or D8.

scholarly journals On finite dual Cayley graphs

2020 ◽  
Vol 18 (1) ◽  
pp. 595-602
Author(s):  
Jiangmin Pan
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Left Regular Representation ◽  
Left Regular ◽  
The Right

Abstract A Cayley graph \Gamma on a group G is called a dual Cayley graph on G if the left regular representation of G is a subgroup of the automorphism group of \Gamma (note that the right regular representation of G is always an automorphism group of \Gamma ). In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible. A few problems worth further research are also proposed.

scholarly journals Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

10.53733/90 ◽  
2021 ◽  
Vol 52 ◽  
pp. 109-143
Author(s):  
Astrid An Huef ◽  
Marcelo Laca ◽  
Iain Raeburn
Keyword(s):  
Regular Representation ◽  
Crossed Product ◽  
Ideal Structure ◽  
Toeplitz Algebra ◽  
Natural Numbers ◽  
Kms States ◽  
C Algebra ◽  
Affine Semigroup ◽  
Natural Dynamics ◽  
The Right

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

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 On the capability of groups

1998 ◽  
Vol 41 (3) ◽  
pp. 487-495 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Automorphism Group ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Integral Homology ◽  
The Third ◽  
Inner Automorphism

We show how the third integral homology of a group plays a role in determining whether a given group is isomorphic to an inner automorphism group. Various necessary conditions, and sufficient conditions, for the existence of such an isomorphism are obtained.

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