scholarly journals Almost-Bieberbach groups with (in)finite outer automorphism group

1998 ◽  
Vol 40 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Wim Malfait ◽  
Andrzej Szczepański
Keyword(s):  
Automorphism Group ◽  
Fundamental Group ◽  
Infinite Order ◽  
Outer Automorphism ◽  
Crystallographic Group ◽  
Outer Automorphism Group ◽  
Outer Automorphisms ◽  
Bieberbach Group ◽  
Interesting Class ◽  
Almost Bieberbach Group

AbstractIf we investigate symmetry of an infra-nilmanifoldM, the outer automorphism group of its fundamental group (an almost-Bieberbach group) is known to be a crucial object. In this paper, we characterise algebraically almost-Bieberbach groupsEwith finite outer automorphism group Out(E). Inspired by the description of Anosov diffeomorphisms onM, we also present an interesting class of infinite order outer automorphisms. Another possible type of infinite order outer automorphisms arises when comparing Out(E) with the outer automorphism group of the underlying crystallographic group ofE.

scholarly journals Crystallographic groups with trivial center and outer automorphism group

2017 ◽  
Vol 164 (2) ◽  
pp. 363-368
Author(s):  
RAFAŁ LUTOWSKI ◽  
ANDRZEJ SZCZEPAŃSKI
Keyword(s):  
Automorphism Group ◽  
Outer Automorphism ◽  
Crystallographic Group ◽  
Crystallographic Groups ◽  
Outer Automorphism Group ◽  
Trivial Center ◽  
Cocompact Subgroup ◽  
Discrete Cocompact Subgroup

AbstractLet Γ be a crystallographic group of dimension n, i.e. a discrete, cocompact subgroup of Isom(ℝn) = O(n) ⋉ ℝn. For any n ⩾ 2, we construct a crystallographic group with a trivial center and trivial outer automorphism group.

Partial *-Automorphisms, Normalizers, and Submodules in Monotone Complete C*-Algebras

2006 ◽  
Vol 58 (6) ◽  
pp. 1144-1202
Author(s):  
Masamichi Hamana
Keyword(s):  
Automorphism Group ◽  
Inverse Semigroup ◽  
Fundamental Group ◽  
Outer Automorphism ◽  
Type Ii ◽  
Partial Isometries ◽  
Von Neumann ◽  
Outer Automorphism Group ◽  
Linear Subspaces ◽  
C Algebras

AbstractFor monotone complete C*-algebras A ⊂ B with A contained in B as a monotone closed C*-subalgebra, the relation X = AsA gives a bijection between the set of all monotone closed linear subspaces X of B such that AX + XA ⊂ X and XX* + X*X ⊂ A and a set of certain partial isometries s in the “normalizer” of A in B, and similarly for the map s ⟼ Ad s between the latter set and a set of certain “partial *-automorphisms” of A. We introduce natural inverse semigroup structures in the set of such X's and the set of partial *-automorphisms of A, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough B the homomorphism becomes surjective and all the partial *-automorphisms of A are realized via partial isometries in B. In particular, the inverse semigroup associated with a type II1 von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the C*-algebra version of these results.

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.

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