CONJUGATION IN TREE AUTOMORPHISM GROUPS

2001 ◽  
Vol 11 (05) ◽  
pp. 529-547 ◽  
Author(s):  
PIOTR W. GAWRON ◽  
VOLODYMYR V. NEKRASHEVYCH ◽  
VITALY I. SUSHCHANSKY
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Rooted Tree ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Rooted Trees ◽  
Full Description ◽  
Finite Tree ◽  
Locally Finite ◽  
Infinite Sequences

It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

scholarly journals Automorphism groups of some variants of lattices

2021 ◽  
Vol 13 (1) ◽  
pp. 142-148
Author(s):  
O.G. Ganyushkin ◽  
O.O. Desiateryk
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Wreath Product ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Natural Generalization ◽  
Space Lattice ◽  
Finite Vector ◽  
Finite Vector Space ◽  
Finite Set

In this paper we consider variants of the power set and the lattice of subspaces and study automorphism groups of these variants. We obtain irreducible generating sets for variants of subsets of a finite set lattice and subspaces of a finite vector space lattice. We prove that automorphism group of the variant of subsets of a finite set lattice is a wreath product of two symmetric permutation groups such as first of this groups acts on subsets. The automorphism group of the variant of the subspace of a finite vector space lattice is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the automorphism group of subspaces lattice and the second is defined by the certain set of symmetric groups.

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.

scholarly journals Automorphism Groups of Rational Circulant Graphs

10.37236/2039 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Mikhail Klin ◽  
István Kovács
Keyword(s):  
Orthogonal Group ◽  
Automorphism Groups ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Circulant Graphs ◽  
Cyclic Groups ◽  
Block Structures

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.

scholarly journals On an Identity for the Cycle Indices of Rooted Tree Automorphism Groups

10.37236/1152 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Stephan G. Wagner
Keyword(s):  
Symmetric Group ◽  
Elementary Proof ◽  
Automorphism Groups ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Cycle Index ◽  
Spanning Forest ◽  
Cycle Indices

This note deals with a formula due to G. Labelle for the summed cycle indices of all rooted trees, which resembles the well-known formula for the cycle index of the symmetric group in some way. An elementary proof is provided as well as some immediate corollaries and applications, in particular a new application to the enumeration of $k$-decomposable trees. A tree is called $k$-decomposable in this context if it has a spanning forest whose components are all of size $k$.

scholarly journals Automorphism Groups of a Class of Cubic Cayley Graphs on Symmetric Groups

2017 ◽  
Vol 24 (04) ◽  
pp. 541-550
Author(s):  
Xueyi Huang ◽  
Qiongxiang Huang ◽  
Lu Lu
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Full Automorphism Group ◽  
Normal Cayley Graph ◽  
The Right

Let Sndenote the symmetric group of degree n with n ≥ 3, S = { cn= (1 2 ⋯ n), [Formula: see text], (1 2)} and Γn= Cay(Sn, S) be the Cayley graph on Snwith respect to S. In this paper, we show that Γn(n ≥ 13) is a normal Cayley graph, and that the full automorphism group of Γnis equal to Aut(Γn) = R(Sn) ⋊ 〈Inn(ϕ) ≅ Sn× ℤ2, where R(Sn) is the right regular representation of Sn, ϕ = (1 2)(3 n)(4 n−1)(5 n−2) ⋯ (∊ Sn), and Inn(ϕ) is the inner isomorphism of Sninduced by ϕ.

scholarly journals CHABAUTY LIMITS OF SIMPLE GROUPS ACTING ON TREES

2018 ◽  
Vol 19 (4) ◽  
pp. 1093-1120
Author(s):  
Pierre-Emmanuel Caprace ◽  
Nicolas Radu
Keyword(s):  
Finite Index ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Simple Groups ◽  
Finite Tree ◽  
Hausdorff Topology ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Bounded Number ◽  
Groups Acting On Trees

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

scholarly journals Automorphism group of the variant of the lattice of partitions of a finite set

2020 ◽  
pp. 115-119
Author(s):  
O. G. Ganyushkin ◽  
O. O. Desiateryk
Keyword(s):  
Automorphism Group ◽  
Wreath Product ◽  
Direct Product ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Natural Generalization ◽  
Generating Sets ◽  
Generalized Wreath Product ◽  
Finite Set

In this paper we consider variants of the lattice of partitions of a finite set and study automorphism groups of this variants. We obtain irreducible generating sets for of the lattice of partitions of a finite set. We prove that the automorphism group of the variant of the lattice of partitions of a finite set is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the direct product of the wreaths products, such that depends on the type of the variant generating partition and the second is defined by the certain set of symmetric groups.

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.

scholarly journals Dense locally finite subgroups of automorphism groups of ultraextensive spaces

2021 ◽  
Vol 391 ◽  
pp. 107966
Author(s):  
Mahmood Etedadialiabadi ◽  
Su Gao ◽  
François Le Maître ◽  
Julien Melleray
Keyword(s):  
Automorphism Groups ◽  
Locally Finite ◽  
Finite Subgroups
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