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 Bounds on finite quasiprimitive permutation groups

2001 ◽  
Vol 71 (2) ◽  
pp. 243-258 ◽  
Author(s):  
Cheryl E. Praeger ◽  
Aner Shalev
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Minimum Degree ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Base Size ◽  
Nontrivial Normal Subgroup

AbstractA permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

ℵ0-categorical structures with arbitrarily fast growth of algebraic closure

2002 ◽  
Vol 67 (3) ◽  
pp. 897-909
Author(s):  
David M. Evans ◽  
M. E. Pantano
Keyword(s):  
Growth Rate ◽  
Automorphism Group ◽  
Finite Subset ◽  
Growth Rates ◽  
Algebraic Closure ◽  
Fast Growth ◽  
Permutation Groups ◽  
Natural Numbers ◽  
Categorical Structure ◽  
Image Position

Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an ℵ0-categorical structure on the set of unordered n-subsets or on the set of n-tuples of elements of . (Recall that by the Ryll-Nardzewski Theorem, if is countable and ℵ0-categorical, the number of the orbits of its automorphism group Aut() on the set of n-tuples from is finite and equals the number of complete n-types consistent with the theory of .) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of ‘folklore’) is that for any sequence (Kn)n∈ℕ of natural numbers there is a countable ℵ0-categorical structure such that the number of orbits of Aut() on the set of n-tuples from is greater than kn for all n.These investigations suggested the study of the growth rate of another sequence. Let be an ℵ0-categorical structure and X be a finite subset of . Let acl(X) be the algebraic closure of X, that is, the union of the finite X-definable subsets of . Equivalently, this is the union of the finite orbits on of Aut()(X), the pointwise stabiliser of X in Aut(). Define

scholarly journals The theorem of Marggraff on primitive permutation groups which contain a cycle

1976 ◽  
Vol 15 (1) ◽  
pp. 125-128 ◽  
Author(s):  
Richard Levingston ◽  
D.E. Taylor
Keyword(s):  
Permutation Group ◽  
Elementary Proof ◽  
Permutation Groups ◽  
Primitive Permutation Group

A short elementary proof is given of the theorem of Marggraff which states that a primitive permutation group which contains a cycle fixing k points is (k+1)-fold transitive. It is then shown that the method of proof actually yields a generalization of Marggraff's theorem.

SUBORBITS IN INFINITE PRIMITIVE PERMUTATION GROUPS

2001 ◽  
Vol 33 (5) ◽  
pp. 583-590 ◽  
Author(s):  
DAVID M. EVANS
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Infinite Cardinal

For every infinite cardinal κ, we construct a primitive permutation group which has a finite suborbit paired with a suborbit of size κ. This answers a question of Peter M. Neumann.

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.

scholarly journals Coprime subdegrees of twisted wreath permutation groups

2019 ◽  
Vol 62 (4) ◽  
pp. 1137-1162
Author(s):  
Alexander Y. Chua ◽  
Michael Giudici ◽  
Luke Morgan
Keyword(s):  
Simple Group ◽  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Primitive Groups ◽  
Group A ◽  
Full Classification

AbstractDolfi, Guralnick, Praeger and Spiga asked whether there exist infinitely many primitive groups of twisted wreath type with non-trivial coprime subdegrees. Here, we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with non-trivial coprime subdegrees. In particular, we define a primitive twisted wreath group G(m, q) constructed from the non-abelian simple group PSL(2, q) and a primitive permutation group of diagonal type with socle PSL(2, q)m, and determine many subdegrees for this group. A consequence is that we determine all values of m and q for which G(m, q) has non-trivial coprime subdegrees. In the case where m = 2 and $q\notin \{7,11,29\}$, we obtain a full classification of all pairs of non-trivial coprime subdegrees.

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.

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).

scholarly journals Extensions of a theorem of Jordan on primitive permutation groups

1983 ◽  
Vol 34 (2) ◽  
pp. 155-171 ◽  
Author(s):  
Martin W. Liebeck
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Finite Degree ◽  
Important Theorem ◽  
Set Of Points

AbstractLet G be a primitive permutation group of finite degree n containing a subgroup H which fixes k points and has r orbits on Δ, the set of points it moves. An old and important theorem of Jordan says that if r = 1 and k ≥ 1 then G is 2-transitive; moreover if H acts primitively on Δ then G is (k + 1)-transitive. Three extensions of this result are proved here: (i) if r = 2 and k ≥ 2 then G is 2-transitive, (ii) if r = 2, n > 9 and H acts primitively on both of its two nontrivial orbits then G is k-primitive, (iii) if r = 3, n > 13 and H acts primitively on each of its three nontrivial orbits, all of which have size at least 3, then G is (k − 1)-primitive.

scholarly journals Primitive permutation groups with a doubly transitive subconstituent

1988 ◽  
Vol 45 (1) ◽  
pp. 66-77 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Normal Subgroup ◽  
Simple Group ◽  
Permutation Group ◽  
Minimal Normal Subgroup ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Diagonal Action ◽  
Product Action ◽  
Finite Set ◽  
Point Stabilizer

AbstractLet Gbe a primitive permutation group on a finite set Ω. We investigate the subconstitutents of G, that is the permutation groups induced by a point stabilizer on its orbits in Ω, in the cases where Ghas a diagonal action or a product action on Ω. In particular we show in these cases that no subconstituent is doubly transitive. Thus if G has a doubly transitive subconstituent we show that G has a unique minimal normal subgroup N and either N is a nonabelian simple group or N acts regularly on Ω: we investigate further the case where N is regular on Ω.

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