scholarly journals Extensions of homomorphisms between localities

2021 ◽  
Vol 9 ◽  
Author(s):  
Ellen Henke
Keyword(s):  
Automorphism Group ◽  
General Definition ◽  
Fusion System ◽  
Normal Subgroups ◽  
General Lemma

Abstract We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal {F}$ depends only on $\mathcal {F}$ as long as the object set of the linking system is $\mathrm {Aut}(\mathcal {F})$ -invariant. This was known to be true for linking systems in Oliver’s definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this article. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.

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 AUTOMORPHISMS OF NONSELFADJOINT DIRECTED GRAPH OPERATOR ALGEBRAS

2009 ◽  
Vol 87 (2) ◽  
pp. 175-196
Author(s):  
BENTON L. DUNCAN
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Maximal Ideal ◽  
Operator Algebras ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Normal Subgroups ◽  
Inner Automorphisms

AbstractWe analyze the automorphism group for the norm closed quiver algebras 𝒯+(Q). We begin by focusing on two normal subgroups of the automorphism group which are characterized by their actions on the maximal ideal space of 𝒯+(Q). To further discuss arbitrary automorphisms we factor automorphism through subalgebras for which the automorphism group can be better understood. This allows us to classify a large number of noninner automorphisms. We suggest a candidate for the group of inner automorphisms.

The abelianization of the congruence IA-automorphism group of a free group

2007 ◽  
Vol 142 (2) ◽  
pp. 239-248 ◽  
Author(s):  
TAKAO SATOH
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Quotient Group ◽  
Homology Group ◽  
Integral Homology ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Homology Groups ◽  
Inner Automorphism

AbstractWe consider the abelianizations of some normal subgroups of the automorphism group of a finitely generated free group. Let Fn be a free group of rank n. For d ≥ 2, we consider a group consisting the automorphisms of Fn which act trivially on the first homology group of Fn with ${\mathbf Z}$/d${\mathbf Z}$-coefficients. We call it the congruence IA-automorphism group of level d and denote it by IAn,d. Let IOn,d be the quotient group of the congruence IA-automorphism group of level d by the inner automorphism group of a free group. We determine the abelianization of IAn,d and IOn,d for n ≥ 2 and d ≥ 2. Furthermore, for n=2 and odd prime p, we compute the integral homology groups of IA2,p for any dimension.

A note on automorphism groups of symmetric cubic graphs

2020 ◽  
pp. 2250018
Author(s):  
Jicheng Ma
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Simple Group ◽  
Minimal Normal Subgroup ◽  
Automorphism Groups ◽  
Cubic Graphs ◽  
Normal Subgroups ◽  
Interesting Consequence ◽  
Cubic Graph

We study [Formula: see text]-arc-transitive cubic graph [Formula: see text], and give a characterization of minimal normal subgroups of the automorphism group. In particular, each [Formula: see text] with quasi-primitive automorphism group is characterized. An interesting consequence of this characterization states that a non-solvable minimal normal subgroup [Formula: see text] contains at most 2 copies of non-abelian simple group when it acts transitively on arcs, or contains at most 6 copies of non-abelian simple group when it acts regularly on vertices.

scholarly journals On normal subgroups of direct products

1990 ◽  
Vol 33 (2) ◽  
pp. 309-319 ◽  
Author(s):  
F. E. A. Johnson
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Finite Number ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Direct Products ◽  
Normal Subgroups ◽  
Free Abelian Groups ◽  
Subdirect Products ◽  
Number Of Classes

We investigate the equivalence classes of normal subdirect products of a product of free groups Fn1 × … × Fnk under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when k≧3, and list the finite number of classes when k = 2.

scholarly journals Maximal normal subgroups of the integral linear group of countable degree

1976 ◽  
Vol 15 (3) ◽  
pp. 439-451 ◽  
Author(s):  
R.G. Burns ◽  
I.H. Farouqi
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Normal Subgroup ◽  
Vector Space ◽  
Free Abelian Group ◽  
Normal Structure ◽  
Normal Subgroups ◽  
Finite Degree ◽  
Infinite Rank ◽  
Countably Infinite

This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

scholarly journals Central automorphisms of finite groups

1986 ◽  
Vol 34 (2) ◽  
pp. 191-198 ◽  
Author(s):  
M. J. Curran ◽  
D. J. McCaughan
Keyword(s):  
Automorphism Group ◽  
Recent Result ◽  
Finite Groups ◽  
General Problem ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Central Automorphism ◽  
Group A ◽  
Central Automorphisms

This paper considers an aspect of the general problem of how the structure of a group influences the structure of its automorphisms group. A recent result of Beisiegel shows that if P is a p-group then the central automorphisms group of P has no normal subgroups of order prime to p. So, roughly speaking, most of the central automorphisms are of p-power order. This generalizes an old result of Hopkins that if Aut P is abelian (so every automorphisms is central), then Aut P is a p-group.This paper uses a different approach to consider the case when P is a π-group. It is shown that the central automorphism group of P has a normal. π′-subgroup only if P has an abelian direct factor whose automorphism group has such a subgroup.

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