scholarly journals Automorphisms of Right-Angled Coxeter Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-10
Author(s):  
Mauricio Gutierrez ◽  
Anton Kaul
Keyword(s):  
Automorphism Group ◽  
Semidirect Product ◽  
Coxeter Groups ◽  
Sufficient Conditions ◽  
Coxeter System ◽  
Group Extensions ◽  
Semidirect Products ◽  
Mild Conditions

If is a right-angled Coxeter system, then is a semidirect product of the group of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, is a semidirect product of by the quotient . We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence and consequently, all group extensions are trivial.

Semigroup Compactifications of Direct and Semidirect Products

1983 ◽  
Vol 26 (2) ◽  
pp. 233-240 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Classical Result ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Products ◽  
Almost Periodic ◽  
Semidirect Products ◽  
Semigroup Compactifications ◽  
Necessary And Sufficient

AbstractA classical result of I. Glicksberg and K. de Leeuw asserts that the almost periodic compactification of a direct product S × T of abelian semigroups with identity is (canonically isomorphic to) the direct product of the almost periodic compactiflcations of S and T. Some efforts have been made to generalize this result and recently H. D. Junghenn and B. T. Lerner have proved a theorem giving necessary and sufficient conditions for an F-compactification of a semidirect product S⊗σT to be a semidirect product of compactiflcations of S and T. A different such theorem is presented here along with a number of corollaries and examples which illustrate its scope and limitations. Some behaviour that can occur for semidirect products, but not for direct products, is exposed

Semidirect products of central groups and groups with equal uniformities

1982 ◽  
Vol 92 (2) ◽  
pp. 239-241
Author(s):  
R. W. Bagley ◽  
J. S. Yang
Keyword(s):  
Automorphism Group ◽  
Semidirect Product ◽  
Type I ◽  
Type Ii ◽  
Topological Groups ◽  
Semidirect Products ◽  
Inner Automorphisms

Let H and K be topological groups, and let HⓈ K denote the semidirect product determined by a homomorphism (η): H → A(K), where A(K) is the automorphism group of K. In this paper we consider two restricted types of semidirect products. We say that HⓈ K is a semidirect product of type I if η(h) is the identity on Z(K), the centre of K, for each hє H, and of type II if η(H) є I(K), where I(K) is the group of inner automorphisms of K. We obtain conditions under which a type II semidirect product of two groups with equal uniformities has equal uniformities, and conditions under which a type I (hence type II) product of two central groups is central. A group G is central if G/Z(G) is compact, where Z(G) is the centre of G.

On the Automorphism Groups of Dowling Geometries

2002 ◽  
Vol 11 (3) ◽  
pp. 311-321
Author(s):  
GARY K. SCHWARTZ
Keyword(s):  
Automorphism Group ◽  
Semidirect Product ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Dowling Lattices

In ‘Automorphisms of Dowling lattices and related geometries’, J. Bonin constructed the automorphism group A of a Dowling lattice as the image of a certain semidirect product, A = θ(K [rtimes ] H). In this work we find necessary and sufficient conditions for this quotient to be the semidirect product A = θ(K) [rtimes ] θ(H). In addition, we include a construction of A that lends itself to computation more readily than that found in Bonin's work.

Topological Left Amenability of Semidirect Products

1981 ◽  
Vol 24 (1) ◽  
pp. 79-85 ◽  
Author(s):  
H. D. Junghenn
Keyword(s):  
Large Class ◽  
Semidirect Product ◽  
Locally Compact ◽  
Semidirect Products ◽  
Topological Semigroups

AbstractLet S and T be locally compact topological semigroups and a semidirect product. Conditions are determined under which topological left amenability of S and T implies that of , and conversely. The results are used to show that for a large class of semigroups which are neither compact nor groups, various notions of topological left amenability coincide.

scholarly journals The Action of the Weyl Group on the $$E_8$$ Root System

2021 ◽  
Author(s):  
Rosa Winter ◽  
Ronald van Luijk
Keyword(s):  
Automorphism Group ◽  
Weyl Group ◽  
Root System ◽  
Sufficient Conditions ◽  
Maximal Clique ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Colored Graphs ◽  
Necessary And Sufficient ◽  
Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

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.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

Groups with prescribed automorphism group: A clarification

1984 ◽  
Vol 27 (1) ◽  
pp. 59-60
Author(s):  
Derek J. S. Robinson
Keyword(s):  
Automorphism Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group A ◽  
Necessary And Sufficient ◽  
Central Automorphisms ◽  
Semisimple Subgroup

In Theorems 1 and 2 of [] necessary and sufficient conditions were given for a group G to have a finite automorphism group Aut G and a semisimple subgroup of central automorphisms AutcG. Recently it occurred to us, as a result of conversations with Ursula Webb, that these conditions could be stated in a much simpler and clearer form. Our purpose here is to record this reformulation. For an explanation ofterminology and notation we refer the reader to [1].

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

scholarly journals On the reconstruction conjecture for separable graphs

1981 ◽  
Vol 30 (3) ◽  
pp. 307-320 ◽  
Author(s):  
V. Krishnamoorthy ◽  
K. R. Parthasarathy
Keyword(s):  
Automorphism Group ◽  
Sufficient Conditions ◽  
Central Block

AbstractSome sufficient conditions for the reconstructability of separable graphs are given proceeding along the lines suggested by Bondy, Greenwell and Hemminger. It is shown that the structure and automorphism group of a central block plays an important role in the reconstruction.

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