Automorphisms of prime-cube order group

2019 ◽  
Vol 11 (03) ◽  
pp. 1950032
Author(s):  
Yi Wang
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Full Automorphism Group ◽  
Order Group ◽  
Transitive Graphs

A graph is called half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. Let [Formula: see text] be a prime-cube order group. In this paper, we determined the structures of the automorphism groups of [Formula: see text], and the result is used to determine the classification of half-arc-transitive graphs of order [Formula: see text] with valency [Formula: see text].

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 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.

AUTOMORPHISM GROUPS OF SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS

2015 ◽  
Vol 93 (2) ◽  
pp. 238-247
Author(s):  
ZHAOHONG HUANG ◽  
JIANGMIN PAN ◽  
SUYUN DING ◽  
ZHE LIU
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Automorphism Groups ◽  
Composition Factor ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Free Order

Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$-power-free order.

On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups

2008 ◽  
Vol 51 (2) ◽  
pp. 261-282 ◽  
Author(s):  
Karl-Hermann Neeb
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Exact Sequence ◽  
Present Note ◽  
Automorphism Groups ◽  
Roots Of Unity ◽  
Quantum Torus ◽  
Twisted Group Algebra ◽  
Quantum Tori

AbstractAn n-dimensional quantum torus is a twisted group algebra of the group ℤn. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for n = 2 the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

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 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.

scholarly journals Cubic symmetric graphs of order twice an odd prime-power

2006 ◽  
Vol 81 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin Ho Kwak
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Prime Power ◽  
Symmetric Graph ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Symmetric Graphs ◽  
Regular Subgroup

AbstractAn automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

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 Regular dessins uniquely determined by a nilpotent automorphism group

2018 ◽  
Vol 21 (3) ◽  
pp. 397-415 ◽  
Author(s):  
Na-Er Wang ◽  
Roman Nedela ◽  
Kan Hu
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Nilpotency Class ◽  
Isomorphism Classes ◽  
One To One

Abstract It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group G are in one-to-one correspondence with the orbits of the action of {{\mathrm{Aut}}(G)} on the ordered generating pairs of G. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group G and it is called uniquely regular. In this paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is reduced to the classification of finite maximally automorphic p-groups G, i.e., the order of the automorphism group of G attains Hall’s upper bound. Maximally automorphic p-groups of nilpotency class three are classified.

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