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.

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 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].

Finite Quotients of the Automorphism Group of a Free Group

1977 ◽  
Vol 29 (3) ◽  
pp. 541-551 ◽  
Author(s):  
Robert Gilman
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Outer Automorphism ◽  
Permutation Representation ◽  
Finitely Generated ◽  
Outer Automorphism Group ◽  
Finite Quotients

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is a free 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.

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