超特殊Z-群的自同构群

确定了超特殊Z-群的自同构群.设G是超特殊Z-群,即G={(1 α_1 α_2···α_n α_(n+1) 0 1 0···0 α_(n+2) ···0 0 0 ··· 0 α_2n 0 0 0··· 1 α_(2n+1) 0 0 0···1 α_(2n+1) 0 0 0···0 1)|α_j∈Z,j=1,2,3,...,2n+1}Aut_cG是AutG中平凡作用在ζG上的自同构形成的正规子群,则AutG=Aut_cG×Z_2,且1→Z···Z}2N→Aut_cG→Sp(2n,Z)→1是正合列.     

The Automorphism Group of a Free-by-Cyclic Group in Rank 2

Let φ be an automorphism of a free group F_n of rank n, and let M_φ = F_n ?_φ ? be the corresponding mapping torus of φ. We study the group Out(M_φ) under certain technical conditions on φ. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of φ in GL₂(?). As an application, we solve the isomorphism problem for the family of F₂-by-? groups, in terms of the two defining automorphisms.     

The Automorphism Tower of a Free Group

It is proved that the automorphism group of any non-abelianfree group F is complete. The key technical step in the proofis that the set of all conjugations by powers of primitive elementsis first-order parameter-free definable in the group Aut(F).     

The Fixed Point Group of a Free Group Automorphism

It is proved that the fixed point group of an arbitrary automorphism of a free group of finite rank has an algorithmically computable basis.     

p6阶群的自同构群的阶

In this paper, we determine the order of automorphism group of p-groups in the third family (Φ3 ) and the fourth family (Φ4 ) in [ 1 ], whose order is p^6 ( p ≥ 3). Here p denotes an odd prime.     

Linear Representations of the Automorphism Group of a Free Group

Let F _n be the free group on n ≥ 2 elements and Aut(F _n ) its group of automorphisms. In this paper we present a rich collection of linear representations of Aut(F _n ) arising through the action of finite-index subgroups of it on relation modules of finite quotient groups of F _n . We show (under certain conditions) that the images of our representations are arithmetic groups. Received: November 2006, Accepted: March 2007     

On the Geometry of the Automorphism Group of a Free Group

The groups Aut(F₃) and Out(F₃) satisfy strictly exponentialisoperimetric inequalities; in particular, they are not automatic.For n 3, Aut (F_n) and Out (F_n) do not admit bounded bicombingsof sub-exponential length, hence they cannot act properly andcocompactly by isometries on any simply-connected space of non-positivecurvature, and they are not biautomatic.     

An Ergodic Action of the Outer Automorphism Group of a Free Group

For n > 2, the action of the outer automorphism group of the rank n free group F _n on Hom(F _n , SU(2))/SU(2) is ergodic with respect to the Lebesgue measure class. The author gratefully acknowledges support from National Science Foundation grants DMS-0405605 and DMS-0103889. Received: September 2005 Revision: January 2006 Accepted: March 2006     

Set Theory is Interpretable in the Automorphism Group of an Infinitely Generated Free Group

In [6] S. Shelah showed that in the endomorphism semi-groupof an infinitely generated algebra which is free in a varietyone can interpret some set theory. It follows from his resultsthat, for an algebra F which is free of infinite rank in avariety of algebras in a language L, if > |L|, then thefirst-order theory of the endomorphism semi-group of F, Th(End(F)),syntactically interprets Th(,L₂), the second-order theory ofthe cardinal . This means that for any second-order sentence of empty language there exists *, a first-order sentence ofsemi-group language, such that for any infinite cardinal >|L|, Th(,L₂)*Th(End(F)) In his paper Shelah notes that it is natural to study a similarproblem for automorphism groups instead of endomorphism semi-groups;a priori the expressive power of the first-order logic for automorphismgroups is less than the one for endomorphism semi-groups. Forinstance, according to Shelah's results on permutation groups[4, 5], one cannot interpret set theory by means of first-orderlogic in the permutation group of an infinite set, the automorphismgroup of an algebra in empty language. On the other hand, onecan do this in the endomorphism semi-group of such an algebra. In [7, 8] the author found a solution for the case of the varietyof vector spaces over a fixed field. If V is a vector spaceof an infinite dimension over a division ring D, then the theoryTh(, L₂) is interpretable in the first-order theory of GL(V),the automorphism group of V. When a field D is countable anddefinable up to isomorphism by a second-order sentence, thenthe theories Th(GL(V)) and Th(, L₂) are mutually syntacticallyinterpretable. In the general case, the formulation is a bitmore complicated. The main result of this paper states that a similar result holdsfor the variety of all groups.     

Embedding the Outer Automorphism Group Out(F n ) of a Free Group of Rank n in the Group Out(F m ) for m>n

It is proved that for every n 1, the group Out(F _n )is embedded in the group Out(F _m ) with m=1+(n-1)k ⁿ , where k is an arbitrary natural number coprime to n-1.     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.     

Rank Two Integral Representations of the Infinite Dihedral Group

We give an effective classification of the representations of the infinite dihedral group in GL ₂(R) where R is either the valuation ring ?_(p) or the ring of p-adic integers.     

A Presentation of the Automorphism Group of the Two-Generator Free Metabelian and Nilpotent Group of Class c

We determine the structure of IA(G)/Inn(G) by giving a set of generators, and showing that IA(G)/Inn(G) is a free abelian group of rank (c – 2)(c + 3)/2. Here G = M ₂, c = x, y, c 2, is the free metabelian nilpotent group of class c.     

On the Automorphism Group of a Free Pro-l Meta-abelian Group and an Application to Galois Representations

In § l of this article, we study group-theoretical properties of some automorphism group Ψ^* of the meta-abelian quotient § of a free pro-l group § of rank two, and show that the conjugacy class of some element of order two of Ψ^* is not determined by the action induced on the abelian quotient § of § in the case of § = 2. In § 2 we apply the results to the outer Galois representation § attached to the curve C deleted one point from an elliptic curve E, and give an example that §_c does not factor through the l-adic representation attached to E.     

二秩无扭群的自同构群和只有两个自同构的二秩无扭群

本文利用Ｋｕｒǒｓ不变量理论和不定方程理论，讨论了二秩无扭群的自同构群，以及有零高元的二秩无扭群只有两个自同构的充要条件．     

A Countable Structure Does Not Have a Free Uncountable Automorphism Group

It is proved that the automorphism group of a countable structurecannot be a free uncountable group. The idea is that insteadof proving that every countable set of equations of a certainform has a solution, it is proved that this holds for a co-meagrefamily of appropriate countable sets of equations. This researchwas partially supported by the Israel Science Foundation.     

Test Elements and Test Rank of a Free Metabelian Group

Siberian Mathematical Journal -     

The Spectrum and Automorphism Group of the Set-Inclusion Graph

Let n,k and l be integers with 1 ≤ k ＜ l ≤ n-1.The set-inclusion graph G(n,k,l) is the graph whose vertex set consists of all k-andl-subsets of[n]={1,2,...,n},where two distinct vertices are adjacent if one of them is contained in the other.In this paper,we determine the spectrum and automorphism group of G(n,k,l).     

The Structure of the (Outer) Automorphism Group of a Bieberbach Group

Referring to Tits alternative, we develop a necessary and sufficient condition to decide whether the normalizer of a finite group of integral matrices is polycyclic-by-finite or is containing a non-Abelian free group. This result is of fundamental importance to conclude whether the (outer) automorphism group of a Bieberbach group is polycyclic-by-finite or has a non-cyclic free subgroup.