scholarly journals Decompositions of the automorphism groups of edge-colored graphs into the direct product of permutation groups

2021 ◽  
Author(s):  
Mariusz Grech
Keyword(s):  
Direct Product ◽  
Automorphism Groups ◽  
Permutation Groups ◽  
Colored Graphs

scholarly journals Direct product of automorphism groups of colored graphs

2004 ◽  
Vol 283 (1-3) ◽  
pp. 81-86 ◽  
Author(s):  
Mariusz Grech ◽  
Andrzej Kisielewicz
Keyword(s):  
Direct Product ◽  
Automorphism Groups ◽  
Colored Graphs

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.

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.

scholarly journals Automorphism groups and Picard groups of additive full subcategories

2010 ◽  
Vol 107 (1) ◽  
pp. 5 ◽  
Author(s):  
Naoya Hiramatsu ◽  
Yuji Yoshino
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Full Subcategory ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Picard Group ◽  
Picard Groups

We study category equivalences between additive full subcategories of module categories over commutative rings. And we are able to define the Picard group of additive full subcategories. The aim of this paper is to study the properties of the Picard groups and show that the automorphism group of an additive full subcategory is a semi-direct product of the Picard group with the group of algebra automorphisms of the ring.

scholarly journals Semiregular Automorphisms in Vertex-Transitive Graphs of Order $3p^2$

10.37236/7499 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Dragan Marušič
Keyword(s):  
Fixed Point ◽  
Prime Order ◽  
Automorphism Groups ◽  
Identity Element ◽  
Permutation Groups ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Free Element

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally $2$-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. It is the purpose of this paper to prove that vertex-transitive graphs of order $3p^2$, where $p$ is a prime, contain semiregular automorphisms.

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.

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