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 Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

scholarly journals EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

2012 ◽  
Vol 92 (1) ◽  
pp. 127-136 ◽  
Author(s):  
CHERYL E. PRAEGER ◽  
CSABA SCHNEIDER
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Automorphism Groups ◽  
Wreath Products ◽  
Error Correcting Codes ◽  
Permutation Groups ◽  
Graph Products ◽  
Product Action ◽  
Hamming Graphs

AbstractWe consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

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.

Determination of torsion Abelian groups by their automorphism groups

2003 ◽  
Vol 67 (3) ◽  
pp. 511-519
Author(s):  
P. Schultz ◽  
A. Sebeldin ◽  
A. L. Sylla
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Torsion Group ◽  
Cyclic Component ◽  
Cyclic Groups

An Abelian torsion group is determined by its automorphism group if and only if its locally cyclic component is determined by its automorphism group. We describe the locally cyclic groups that are determined by their automorphism groups.

On orbit closures of groups

2015 ◽  
Vol 14 (09) ◽  
pp. 1540008
Author(s):  
S. B. Mulay
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Permutation Groups ◽  
Arbitrary Group ◽  
Orbit Closures ◽  
Power Set

We establish some results about orbit closures of permutation groups and their relationship to the automorphism groups of power-set graphs with colorings. One of our theorems proves that an arbitrary group can be realized as the automorphism group of a colored power-set graph.

ALL GROUPS ARE OUTER AUTOMORPHISM GROUPS OF SIMPLE GROUPS

2001 ◽  
Vol 64 (3) ◽  
pp. 565-575 ◽  
Author(s):  
MANFRED DROSTE ◽  
MICHÈLE GIRAUDET ◽  
RÜDIGER GÖBEL
Keyword(s):  
Automorphism Group ◽  
Recent Result ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Simple Groups ◽  
Symmetry Groups ◽  
Normal Subgroups ◽  
Outer Automorphism Group ◽  
Relational Structures ◽  
Outer Automorphism Groups

It is shown that each group is the outer automorphism group of a simple group. Surprisingly, the proof is mainly based on the theory of ordered or relational structures and their symmetry groups. By a recent result of Droste and Shelah, any group is the outer automorphism group Out (Aut T) of the automorphism group Aut T of a doubly homogeneous chain (T, [les ]). However, Aut T is never simple. Following recent investigations on automorphism groups of circles, it is possible to turn (T, [les ]) into a circle C such that Out (Aut T) [bcong ] Out (Aut C). The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and the result follows.

scholarly journals Abelian permutation groups with graphical representations

2021 ◽  
Author(s):  
Mariusz Grech ◽  
Andrzej Kisielewicz
Keyword(s):  
Permutation Group ◽  
Automorphism Groups ◽  
Permutation Groups ◽  
Graphical Representations ◽  
Colored Graphs

AbstractIn this paper we characterize those automorphism groups of colored graphs and digraphs that are abelian as abstract groups. This is done in terms of basic permutation group properties. Using Schur’s classical terminology, what we provide is characterizations of the classes of 2-closed and $$2^*$$ 2 ∗ -closed abelian permutation groups. This is the first characterization concerning these classes since they were defined.

Algebras with Transitive Automorphism Groups

1986 ◽  
Vol 29 (2) ◽  
pp. 224-226
Author(s):  
L. G. Sweet ◽  
J. A. MacDougall
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Automorphism Groups ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Transitive Automorphism ◽  
Cyclic Groups ◽  
Finite Dimensional

AbstractLet A be a finite dimensional algebra (not necessarily associative) over a field, whose automorphism group acts transitively. It is shown that K = GF(2) and A is a Kostrikin algebra. The automorphism group is determined to be a semi-direct product of two cyclic groups. The number of such algebras is also calculated.

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.

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