scholarly journals Some finiteness conditions for automorphism groups

1987 ◽  
Vol 29 (2) ◽  
pp. 259-265
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni
Keyword(s):  
Automorphism Group ◽  
Finite Order ◽  
Automorphism Groups ◽  
Torsion Group ◽  
Finite Subgroup ◽  
Finiteness Conditions

Many authors have investigated the behaviour of the elements of finite order of a group G when finiteness conditions are imposed on the automorphism group Aut G of G. The first result was obtained in 1955 by Baer [1], who proved thata torsion group with finitely many automorphisms is finite. This theorem was generalized by Nagrebeckii in [6], where he proved that if the automorphism group Aut G is finite then the set of elements of finite order of G is a finite subgroup.

Automorphism groups and invariant theory on ℙN

2018 ◽  
Vol 17 (09) ◽  
pp. 1850162 ◽  
Author(s):  
João Alberto de Faria ◽  
Benjamin Hutz
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Invariant Theory ◽  
Automorphism Groups ◽  
Finite Subgroup ◽  
Fixed Point Algorithm ◽  
Projective Linear Group ◽  
Rationality Problems ◽  
Stabilizer Group ◽  
A Finite Group

Let [Formula: see text] be a field and [Formula: see text] a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group [Formula: see text]. The group of automorphisms, or stabilizer group, of a given [Formula: see text] for this action is known to be a finite group. In this paper, we apply methods of invariant theory to automorphism groups by addressing two mainly computational problems. First, given a finite subgroup of [Formula: see text], determine endomorphisms of [Formula: see text] with that group as a subgroup of its automorphism group. In particular, we show that every finite subgroup occurs infinitely often and discuss some associated rationality problems. Inversely, given an endomorphism, determine its automorphism group. In particular, we extend the Faber–Manes–Viray fixed-point algorithm for [Formula: see text] to endomorphisms of [Formula: see text]. A key component is an explicit bound on the size of the automorphism group depending on the degree of the endomorphism.

scholarly journals COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS, FREE AMALGAMATION AND AUTOMORPHISM GROUPS

2019 ◽  
Vol 85 (1) ◽  
pp. 199-223 ◽  
Author(s):  
DAOUD SINIORA ◽  
SŁAWOMIR SOLECKI
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Extension Property ◽  
Finite Subgroup ◽  
Homogeneous Structure ◽  
Extension Theorems ◽  
Locally Finite ◽  
Finite Structures ◽  
Image Position ◽  
Fraïssé Class

AbstractWe give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

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.

scholarly journals BOUNDEDNESS PROPERTIES OF AUTOMORPHISM GROUPS OF FORMS OF FLAG VARIETIES

2020 ◽  
Vol 25 (4) ◽  
pp. 1161-1184
Author(s):  
A. GULD
Keyword(s):  
Automorphism Group ◽  
Linear Group ◽  
General Linear Group ◽  
Automorphism Groups ◽  
Finite Subgroup ◽  
Flag Variety ◽  
General Linear ◽  
Roots Of Unity ◽  
Flag Varieties ◽  
Projective General Linear Group

Abstract We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.) Let K be a field of characteristic 0, containing all roots of unity. Let the K-variety X be a form of an admissible flag variety. We prove that X is either ruled, or the automorphism group of X is bounded, meaning that there exists a constant C ∈ ℕ such that if G is a finite subgroup of AutK(X), then the cardinality of G is smaller than C.

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 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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