Strong map-symmetry of SL(3,K) and PSL(3,K) for every finite field K

2020 ◽  
pp. 2150048
Author(s):  
Antonio Breda d’Azevedo ◽  
Domenico A. Catalano
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Automorphism Groups ◽  
Group Automorphism ◽  
Regular Maps ◽  
Strong Map

In this paper, we show that for any finite field [Formula: see text], any pair of map-generators (that is when one of the generators is an involution) of [Formula: see text] and [Formula: see text] has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group [Formula: see text] or [Formula: see text] is reflexible, or equivalently, there are no chiral regular maps with automorphism group [Formula: see text] or [Formula: see text]. As remarked by Leemans and Liebeck, also [Formula: see text] and [Formula: see text] are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

scholarly journals Introducing edge-biregular maps

2021 ◽  
Author(s):  
Olivia Reade
Keyword(s):  
Automorphism Group ◽  
Euler Characteristic ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Triangle Group ◽  
Supporting Surface ◽  
Regular Maps ◽  
Special Cases ◽  
Edge Colourings

AbstractWe introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples of such maps. Edge-biregular maps of type (k, l) occur as smooth normal quotients of a particular index two subgroup of $$T_{k,l}$$ T k , l , the full triangle group describing regular plane (k, l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.

Finite Groups Which are Automorphism Groups of Infinite Groups Only

1985 ◽  
Vol 28 (1) ◽  
pp. 84-90
Author(s):  
Jay Zimmerman
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Schur Multiplier ◽  
Infinite Groups ◽  
Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

scholarly journals FREE AMALGAMATION AND AUTOMORPHISM GROUPS

2016 ◽  
Vol 81 (3) ◽  
pp. 936-947 ◽  
Author(s):  
ANDREAS BAUDISCH
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Lie Algebras ◽  
Automorphism Groups ◽  
Nilpotent Lie Algebras ◽  
Fixed Field

AbstractWe show that the class of graded c-nilpotent Lie algebras over a fixed field K is closed under free amalgamation. In [1] this result was applied, but its proof was incorrect. In case of a finite field K we obtain a Fraïssé limit of all finite graded c-nilpotent Lie algebras over K. This gives an example for the following more general considerations. The existence of free amalgamation for the age of a Fraïssé limit implies the universality of its automorphism group for all automorphism groups of substructures of that Fraïssé limit. We use [6] and [5].

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 On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

scholarly journals On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains

2016 ◽  
Vol 59 (2) ◽  
pp. 346-353
Author(s):  
Steven Krantz
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Holomorphic Functions ◽  
Biholomorphic Equivalence ◽  
Algebras Of Holomorphic Functions

AbstractWe study and generalize a classical theoremof L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group

scholarly journals AUTOMORPHISM GROUPS OF RANDOMIZED STRUCTURES

2017 ◽  
Vol 82 (3) ◽  
pp. 1150-1179 ◽  
Author(s):  
TOMÁS IBARLUCÍA
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Original Structure ◽  
Polish Groups ◽  
Separable Models

AbstractWe study automorphism groups of randomizations of separable structures, with focus on the ℵ0-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the ℵ0-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never ℵ0-categorical (except in basic cases).

scholarly journals A note on subgroups of automorphism groups of full shifts

2016 ◽  
Vol 38 (4) ◽  
pp. 1588-1600 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Endomorphism Monoid ◽  
Direct Products ◽  
Free Products ◽  
Group Case ◽  
Sofic Shift ◽  
Full Shift

We discuss the set of subgroups of the automorphism group of a full shift and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.

scholarly journals A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

2019 ◽  
pp. 729
Author(s):  
Mahsa Mirzargar
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
The Other ◽  
Power Graph ◽  
Commuting Graph ◽  
Vertex Set ◽  
Delta G ◽  
Nite Group

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

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