On the number of graphs for which a given permutation group is the automorphism group

Cybernetics ◽  
1973 ◽  
Vol 6 (6) ◽  
pp. 862-870
Author(s):  
M. Kh. Klin
Keyword(s):  
Automorphism Group ◽  
Permutation Group

scholarly journals Infinite Graphical Frobenius Representations

10.37236/7294 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Mark E. Watkins
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Polynomial Growth ◽  
Free Products ◽  
Locally Finite ◽  
Finitely Generated Groups ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Planar Maps ◽  
Transitive Graphs

A graphical Frobenius representation (GFR) of a Frobenius (permutation) group $G$ is a graph $\Gamma$ whose automorphism group Aut$(\Gamma)$ acts as a Frobenius permutation group on the vertex set of $\Gamma$, that is, Aut$(\Gamma)$ acts vertex-transitively with the property that all nonidentity automorphisms fix either exactly one or zero vertices and there are some of each kind. The set $K$ of all fixed-point-free automorphisms together with the identity is called the kernel of $G$. Whenever $G$ is finite, $K$ is a regular normal subgroup of $G$ (F. G. Frobenius, 1901), in which case $\Gamma$ is a Cayley graph of $K$. The same holds true for all the infinite instances presented here.Infinite, locally finite, vertex-transitive graphs can be classified with respect to (i) the cardinality of their set of ends and (ii) their growth rate. We construct families of infinite GFRs for all possible combinations of these two properties. There exist infinitely many GFRs with polynomial growth of degree $d$ for every positive integer $d$, and there exist infinite families of GFRs of exponential growth, both $1$-ended and infinitely-ended, that underlie infinite chiral planar maps. There also exist GFRs of free products of finitely many finitely generated groups. Graphs of connectivity 1 having a Frobenius automorphism group are characterized.

Automorphism Groups of Algebras of Finite Type

1972 ◽  
Vol 24 (6) ◽  
pp. 1065-1069 ◽  
Author(s):  
Matthew Gould
Keyword(s):  
Automorphism Group ◽  
Finite Number ◽  
Finite Type ◽  
Permutation Group ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈A; F〉 where A and F are nonvoid sets and every element of F is a function, defined on A, of some finite number of variables. Armbrust and Schmidt showed in [1] that for any finite nonvoid set A, every group G of permutations of A is the automorphism group of an algebra defined on A and having only one operation, whose rank is the cardinality of A. In [6], Jónsson gave a necessary and sufficient condition for a given permutation group to be the automorphism group of an algebra, whereupon Plonka [8] modified Jonsson's condition to characterize the automorphism groups of algebras whose operations have ranks not exceeding a prescribed bound.

scholarly journals Reconstruction of homogeneous relational structures

2007 ◽  
Vol 72 (3) ◽  
pp. 792-802 ◽  
Author(s):  
Silvia Barbina ◽  
Dugald Macpherson
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Group Structure ◽  
Baire Category ◽  
Abstract Group ◽  
Relational Structures ◽  
Small Index Property ◽  
Categorical Structure ◽  
Small Index

This paper contains a result on the reconstruction of certain homogeneous transitive ω-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly non-standard sense, coming from Baire category).Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut() of an ω-categorical structure determines the topology on Aut(), and hence determines up to bi-interpretability, by [1]; they can also give conditions under which the abstract group Aut() determines the permutation group ⟨Aut (), ⟩. so determines up to bi-definability. One such condition has been identified by M. Rubin in [12], and it is related to the definability, in Aut(), of point stabilisers. If the condition holds, the structure is said to have a weak ∀∃ interpretation, and Aut() determines up to bi-interpretability or, in some cases, up to bi-definability.A better-known approach to reconstruction is via the ‘small index property’: an ω-categorical stucture has the small index property if any subgroup of Aut() of index less than is open. This guarantees that the abstract group structure of Aut() determines the topology, so if is ω-categorical with Aut() ≅ Aut() then and are bi-interpretable.

The Number of Graphs with a given Automorphism Group

1968 ◽  
Vol 20 ◽  
pp. 1068-1076 ◽  
Author(s):  
J. Sheehan
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Finite Groups ◽  
Representations Of Finite Groups ◽  
Multiple Edges

In this paper, the graphs under consideration may have multiple edges but they do not have loops. We enumerate the number N[H: n, p] of topologically distinct graphs with n vertices and p edges whose automorphism group is the permutation group H. As in (5), this enumeration is considered in the context of the theory of permutation representations of finite groups. We begin with some definitions and notation.

scholarly journals Cyclic Permutation Groups that are Automorphism Groups of Graphs

2019 ◽  
Vol 35 (6) ◽  
pp. 1405-1432 ◽  
Author(s):  
Mariusz Grech ◽  
Andrzej Kisielewicz
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Boolean Functions ◽  
Automorphism Groups ◽  
Cyclic Permutation ◽  
Permutation Groups ◽  
Symmetry Groups ◽  
Cyclic Groups

Abstract In this paper we establish conditions for a permutation group generated by a single permutation to be an automorphism group of a graph. This solves the so called concrete version of König’s problem for the case of cyclic groups. We establish also similar conditions for the symmetry groups of other related structures: digraphs, supergraphs, and boolean functions.

Jordan permutation groups and limits of 𝐷-relations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asma Ibrahim Almazaydeh ◽  
Dugald Macpherson
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Categorical Structure

Abstract We construct via Fraïssé amalgamation an 𝜔-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a “limit of 𝐷-relations”. The construction is based on a semilinear order whose elements are labelled by sets carrying a 𝐷-relation, with strong coherence conditions governing how these 𝐷-sets are inter-related.

scholarly journals On the Existence of Frobenius Digraphical Representations

10.37236/7097 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Pablo Spiga
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Frobenius Group ◽  
Natural Extension ◽  
Partial Answer ◽  
Transitive Permutation Group ◽  
Frobenius Groups ◽  
Frobenius Complement ◽  
Vertex Set

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A digraphical, respectively graphical, Frobenius representation, DFR and GFR for short, of a Frobenius group $F$ is a digraph, respectively graph, whose automorphism group as a group of permutations of the vertex set is $F$. The problem of classifying which Frobenius groups admit a DFR and GFR has been proposed by Mark Watkins and Thomas Tucker and is a natural extension of the problem of classifying which groups that have a digraphical, respectively graphical, regular representation.In this paper, we give a partial answer to a question of Mark Watkins and Thomas Tucker concerning Frobenius representations: "All but finitely many Frobenius groups with a given Frobenius complement have a DFR".  

scholarly journals Solvability of a Class of Rank 3 Permutation Groups

1971 ◽  
Vol 41 ◽  
pp. 89-96 ◽  
Author(s):  
D.G. Higman
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Regular Graph ◽  
Strongly Regular Graph ◽  
Permutation Groups ◽  
Theoretical Interpretation ◽  
Finite Set ◽  
Strongly Regular ◽  
Even Order

1. Introduction. Let G be a rank 3 permutation group of even order on a finite set X, |X| = n, and let Δ and Γ be the two nontrivial orbits of G in X×X under componentwise action. As pointed out by Sims [6], results in [2] can be interpreted as implying that the graph = (X, Δ) is a strongly regular graph, the graph theoretical interpretation of the parameters k, l, λ and μ of [2] being as follows: k is the degree of , λ is the number of triangles containing a given edge, and μ is the number of paths of length 2 joining a given vertex P to each of the l vertices ≠ P which are not adjacent to P. The group G acts as an automorphism group on and on its complement = (X,Γ).

Graphical Regular Representations of Non-Abelian Groups, I

1972 ◽  
Vol 24 (6) ◽  
pp. 993-1008 ◽  
Author(s):  
Lewis A. Nowitz ◽  
Mark E. Watkins
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Abelian Groups ◽  
Regular Representation ◽  
Abstract Group ◽  
Regular Representations ◽  
Graphical Regular Representation ◽  
Vertex Set

In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). IfXis such a graph with vertex setV(X)and automorphism groupA(X),we say thatXis agraphical regular representation(GRR) of a given abstract groupGif(I) G ≅ A(X) , and(II)A(X)acts onV(X) as a regular permutation group; that is, givenu, v∈V(X), there exists a uniqueφ∈A(X)for whichφ(u) =v.That for any abstract groupGthere exists a graphXsatisfying (I) is well-known (cf. [3]).

scholarly journals A report on stable graphs

1973 ◽  
Vol 15 (2) ◽  
pp. 163-171 ◽  
Author(s):  
D. A. Holton
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Complete Graph ◽  
Symmetric Groups ◽  
Vertex Set

It is the aim of this paper to introduce a new concept relating various subgroups of the automorphism group of a graph to corresponding subgraphs. Throughout g will denote a (Michigan) graph on a vertex set V(¦V¦ =n) and Γ(g)=G will be the automorphism group of G considered as a permutation group on V.En, Cn, Dn and Sn are the identity, cyclic, dihedral, and symmetric groups acting on a set of size n, while Sp(q) is the permutation group of pq objects which is isomorphic to Sp but is q-fold in the sense that the objects are permuted q at a time [6]. H ≦ G means that H is a subgroup of G. Other group concepts can be found in Wielandt [7]. The graphs G1 ∪ G2, G1 + G2, G1 × G2, and G1[G2] along with their corresponding groups are as defined in, for example, Harary [4]. Finally we use Kn for the complete graph on n vertices.

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