scholarly journals Automorphism groups of Cayley digraphs of ${\Bbb Z}_p^3$

10.37236/238 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Edward Dobson ◽  
István Kovács
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Full Automorphism Group ◽  
Product Action

We calculate the full automorphism group of Cayley digraphs of ${\Bbb Z}_p^3$, $p$ an odd prime, as well as determine the $2$-closed subgroups of $S_m \wr S_p$ with the product action.

scholarly journals A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2935
Author(s):  
Bo Ling ◽  
Wanting Li ◽  
Bengong Lou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Cayley Graphs ◽  
Vital Role ◽  
Alternating Group ◽  
Full Automorphism Group ◽  
Base Group

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

Jordan property and automorphism groups of normal compact Kähler varieties

2018 ◽  
Vol 20 (03) ◽  
pp. 1750024 ◽  
Author(s):  
Jin Hong Kim
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Projective Varieties ◽  
Full Automorphism Group ◽  
Identity Component ◽  
Arbitrary Dimensions ◽  
Type Decomposition

It has been recently shown by Meng and Zhang that the full automorphism group [Formula: see text] is a Jordan group for all projective varieties in arbitrary dimensions. The aim of this paper is to show that the full automorphism group [Formula: see text] is, in fact, a Jordan group even for all normal compact Kähler varieties in arbitrary dimensions. The meromorphic structure of the identity component of the automorphism group and its Rosenlicht-type decomposition play crucial roles in the proof.

scholarly journals Equality of certain automorphism groups of finite p-groups

2016 ◽  
Vol 15 (03) ◽  
pp. 1650056
Author(s):  
Deepak Gumber ◽  
Hemant Kalra
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Recent Past ◽  
Full Automorphism Group ◽  
Technical Lemma ◽  
Necessary And Sufficient

Let [Formula: see text] be a finite [Formula: see text]-group and let Aut([Formula: see text]) denote the full automorphism group of [Formula: see text]. In the recent past, there has been interest in finding necessary and sufficient conditions on [Formula: see text] such that certain subgroups of Aut([Formula: see text]) are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.

Symmetric designs admitting flag-transitive and point-primitive almost simple automorphism groups of Lie type

2017 ◽  
Vol 16 (10) ◽  
pp. 1750192
Author(s):  
Yajie Wang ◽  
Shenglin Zhou
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Symmetric Design ◽  
Symmetric Designs ◽  
Full Automorphism Group ◽  
Design Formula ◽  
Groups Of Lie Type

Let [Formula: see text] be a subgroup of the full automorphism group of a [Formula: see text]-[Formula: see text] symmetric design [Formula: see text]. If [Formula: see text] is flag-transitive and point-primitive, then Soc[Formula: see text] cannot be [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

scholarly journals On plane curves given by separated polynomials and their automorphisms

2020 ◽  
Vol 20 (1) ◽  
pp. 61-70
Author(s):  
Matteo Bonini ◽  
Maria Montanucci ◽  
Giovanni Zini
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Algebraic Closure ◽  
Sufficient Conditions ◽  
Plane Curve ◽  
Plane Curves ◽  
Prime Field ◽  
Full Automorphism Group ◽  
Ag Codes ◽  
Finite Prime Field

AbstractLet 𝓒 be a plane curve defined over the algebraic closure K of a finite prime field 𝔽p by a separated polynomial, that is 𝓒 : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of 𝓒 when m ≢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of 𝓒 to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve 𝓒 : X(qr – 1)/(q–1) = Yqr–1 + Yqr–2 + … + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.

scholarly journals Automorphisms and Enumeration of Switching Classes of Tournaments.

10.37236/1516 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
L. Babai ◽  
P. J. Cameron
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Probability Distributions ◽  
Explicit Construction ◽  
Linear Orders ◽  
Full Automorphism Group ◽  
Odd Order ◽  
Vertex Set ◽  
Countably Infinite ◽  
Switching Class

Two tournaments $T_1$ and $T_2$ on the same vertex set $X$ are said to be switching equivalent if $X$ has a subset $Y$ such that $T_2$ arises from $T_1$ by switching all arcs between $Y$ and its complement $X\setminus Y$. The main result of this paper is a characterisation of the abstract finite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral. Moreover, if $G$ is such a group, then there is a switching class $C$, with Aut$(C)\cong G$, such that every subgroup of $G$ of odd order is the full automorphism group of some tournament in $C$. Unlike previous results of this type, we do not give an explicit construction, but only an existence proof. The proof follows as a special case of a result on the full automorphism group of random $G$-invariant digraphs selected from a certain class of probability distributions. We also show that a permutation group $G$, acting on a set $X$, is contained in the automorphism group of some switching class of tournaments with vertex set $X$ if and only if the Sylow 2-subgroups of $G$ are cyclic or dihedral and act semiregularly on $X$. Applying this result to individual permutations leads to an enumeration of switching classes, of switching classes admitting odd permutations, and of tournaments in a switching class. We conclude by remarking that both the class of switching classes of finite tournaments, and the class of "local orders" (that is, tournaments switching-equivalent to linear orders), give rise to countably infinite structures with interesting automorphism groups (by a theorem of Fraïssé).

scholarly journals Finitely generated groups with virtually free automorphism groups

1995 ◽  
Vol 38 (3) ◽  
pp. 475-484 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Finitely Generated ◽  
Full Automorphism Group ◽  
Finitely Generated Groups ◽  
Special Cases ◽  
Central Quotient ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

It is shown that the full automorphism group of a finitely generated group G is virtually free if and only if the center Z(G) is finitely generated of torsion-free rank r at most two and, depending on the value of r, the central quotient G/Z(G) belongs to one of three precisely defined classes of virtually free groups. Some consequences and special cases are also discussed.

Automorphism Groups of Bouwer Graphs

2018 ◽  
Vol 25 (03) ◽  
pp. 493-508 ◽  
Author(s):  
Mimi Zhang ◽  
Jinxin Zhou
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Transitive Graph ◽  
Full Automorphism Group ◽  
Vertex Set ◽  
Edge Transitive ◽  
Positive Integers

Let k, m and n be three positive integers such that 2m ≡ 1 (mod n) and k ≥ 2. The Bouwer graph, which is denoted by B(k, m, n), is the graph with vertex set [Formula: see text] and two vertices being adjacent if they can be written as (a, b) and (a + 1, c), where either c = b or [Formula: see text] differs from [Formula: see text] in exactly one position, say the jth position, where [Formula: see text]. Every B(k, m, n) is a vertex- and edge-transitive graph, and Bouwer proved that B(k, 6, 9) is half-arc-transitive for every k ≥ 2. In 2016, Conder and Žitnik gave the classification of half-arc-transitive Bouwer graphs. In this paper, the full automorphism group of every B(k, m, n) is determined.

scholarly journals Line-transitive Automorphism Groups of Linear Spaces

10.37236/1227 ◽  
1995 ◽  
Vol 3 (1) ◽  
Author(s):  
Alan R Camina ◽  
Susanne Mischke
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Linear Space ◽  
Automorphism Groups ◽  
Linear Spaces ◽  
Full Automorphism Group ◽  
Transitive Automorphism

In this paper we prove the following theorem. Let $\cal S$ be a linear space. Assume that $\cal S$ has an automorphism group $G$ which is line-transitive and point-imprimitive with $k < 9$. Then $\cal S$ is one of the following:- (a) A projective plane of order $4$ or $7$, (b) One of $2$ linear spaces with $v=91$ and $k=6$, (c) One of $467$ linear spaces with $v=729$ and $k=8$. In all cases the full automorphism group Aut(${\cal S} \!$) is known.

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.

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