ALL GROUPS ARE OUTER AUTOMORPHISM GROUPS OF SIMPLE GROUPS

2001 ◽  
Vol 64 (3) ◽  
pp. 565-575 ◽  
Author(s):  
MANFRED DROSTE ◽  
MICHÈLE GIRAUDET ◽  
RÜDIGER GÖBEL
Keyword(s):  
Automorphism Group ◽  
Recent Result ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Simple Groups ◽  
Symmetry Groups ◽  
Normal Subgroups ◽  
Outer Automorphism Group ◽  
Relational Structures ◽  
Outer Automorphism Groups

It is shown that each group is the outer automorphism group of a simple group. Surprisingly, the proof is mainly based on the theory of ordered or relational structures and their symmetry groups. By a recent result of Droste and Shelah, any group is the outer automorphism group Out (Aut T) of the automorphism group Aut T of a doubly homogeneous chain (T, [les ]). However, Aut T is never simple. Following recent investigations on automorphism groups of circles, it is possible to turn (T, [les ]) into a circle C such that Out (Aut T) [bcong ] Out (Aut C). The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and the result follows.

scholarly journals Cannon–Thurston maps for $${{\,\mathrm{CAT}\,}}(0)$$ groups with isolated flats

2021 ◽  
Author(s):  
Beeker Benjamin ◽  
Matthew Cordes ◽  
Giles Gardam ◽  
Radhika Gupta ◽  
Emily Stark
Keyword(s):  
Automorphism Groups ◽  
Structure Theorem ◽  
Hyperbolic Group ◽  
Outer Automorphism ◽  
Hyperbolic Groups ◽  
Normal Subgroups ◽  
Infinite Index ◽  
Outer Automorphism Groups ◽  
Visual Boundaries

AbstractMahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group (Mitra in Topology, 37(3):527–538, 1998). We prove that Cannon–Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite infinite-index normal $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) subgroups with isolated flats in non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal $$\mathbb {Z}^2$$ Z 2 subgroups.

SET IDEALS WITH ISOMORPHIC SYMMETRY GROUPS

2002 ◽  
Vol 34 (1) ◽  
pp. 37-45
Author(s):  
PETER BIRYUKOV ◽  
VALERY MISHKIN
Keyword(s):  
Automorphism Group ◽  
Symmetry Group ◽  
Wide Class ◽  
Metric Spaces ◽  
Outer Automorphism ◽  
Symmetry Groups ◽  
Outer Automorphism Group ◽  
Complete Symmetry ◽  
Thin Sets ◽  
Complete Symmetry Group

A criterion of isomorphism for symmetry groups of set ideals is provided in terms of ideal quotients and cardinal invariants. Furthermore, set ideals with complete symmetry group are characterized. They form a wide class, comprising, for example, all uniform dense ideals and the ideals of ‘thin’ sets in separable metric spaces. If the symmetry group of a set ideal is not complete, then its outer automorphism group is shown to be cyclic of order 2.

Recovering ordered structures from quotients of their automorphism groups

2003 ◽  
Vol 68 (4) ◽  
pp. 1189-1198
Author(s):  
M. Giraudet ◽  
J. K. Truss
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Structures ◽  
Outer Automorphism Group ◽  
Simple Lattice

AbstractWe show that the ‘tail’ of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

scholarly journals Continuous trace C*-algebras with given Dixmer-Douady class

1985 ◽  
Vol 38 (3) ◽  
pp. 394-407 ◽  
Author(s):  
Iain Raeburn ◽  
Joseph L. Taylor
Keyword(s):  
Automorphism Group ◽  
Outer Automorphism ◽  
Explicit Construction ◽  
Irreducible Representations ◽  
Outer Automorphism Group ◽  
C Algebras ◽  
Finite Dimensional ◽  
Continuous Trace ◽  
Separable Algebras

AbstractWe give an explicit construction of a continuous trace C*algebra with prescribed Dixmier-Douady class, and with only finite-dimensional irreducible representations. These algebras often have non-trivial automorphisms, and we show how a recent description of the outer automorphism group of a stable continuous trace C*algebra follows easily from our main result. Since our motivation came from work on a new notion of central separable algebras, we explore the connections between this purely algebraic subject and C*a1gebras.

Quasirecognition of E6(q) by the orders of maximal abelian subgroups

2018 ◽  
Vol 17 (07) ◽  
pp. 1850122 ◽  
Author(s):  
Zahra Momen ◽  
Behrooz Khosravi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Simple Group ◽  
Outer Automorphism ◽  
Composition Factor ◽  
Outer Automorphism Group ◽  
Abelian Subgroups ◽  
A Finite Group ◽  
Nonabelian Composition Factor

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

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