Dynamics of free group automorphisms

A generating set for the automorphism group of a graph product of abelian groups

We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined in [7]. We follow closely the plan of M. Laurence's paper [11].

Smallest non-cyclic quotients of the automorphism group of free groups

We give an new, elementary proof of the result that the smallest non-cyclic quotients of automorphism group of free group is the linear group over the field of two elements, and moreover all minimal quotients are obtained by the standard projection composed with an automorphism of the image. This result, originally due to Baumeister-Kielak-Pierro, proves a conjecture of Mecchia-Zimmermann.

Automorphisms of pure braid Groups

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to…

Tits Alternative for the 3-dimensional tame automorphism group

We prove the strong Tits alternative for the 3-dimensional tame automorphism group over a field K of characteristic zero.

The automorphism group of a nonsplit metacyclic 2-group

We determine the structure of automorphism group or each nonsplit metacyclic 2-group. This completes the work on automorphism groups of metacyclic $p$-groups.

The automorphism group of a multi-GGS group

A multi-GGS-group is a group of automorphisms of a regular rooted tree, generalising the Gupta--Sidki $p$-groups. We compute the automorphism groups of all non-constant multi-GGS-groups.

Automorphisms of two-dimensional RAAGs and partially symmetric automorphisms of free groups

We compute the virtual cohomological dimension (VCD) of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a two-dimensional right-angled Artin group. In the case of a right-angled group with defining graph a tree, the bounds agree.

On embeddings between outer automorphism groups of right-angled Artin groups

Using abelian coverings of Salvetti complexes, embeddings of outer automorphism groups of right-angled Artin groups (RAAGs) into outer automorphism groups of their particular characteristic subgroups are constructed. Virtual embeddings of outer automorphism groups of finitely generated groups having the unique root property into outer automorphism groups of their particular subgroups are also given. These results provide us with rich examples of (virtual) embeddings between outer automorphism groups of RAAGs. Finite subgroups of pure (outer) automorphism groups of RAAGs are also investigated. …

Automorphism groups of $2$-groups of coclass at most $3$

Automorphism groups of $2$-groups of coclass at most $3$ are investigated.

Class-preserving automorphisms of universal hyperlinear groups

We show that the group of class-preserving automorphisms of a universal hyperlinear group has index 2 inside the group of all automorphisms.

一些经典群的局部自同构

Local automorphisms of some classical groups

研究经典群的局部自同构与自同构的关系，发现部分情况下局部自同构即为自同构，其余情况则与自同构密切相关。

On the polynomial automorphisms of a group

We prove that if a group is nilpotent (resp. metabelian), then so is the subgroup of its automorphism group generated by all polynomial automorphisms.

The Automorphism Groups of the Groups of Order 8p^2

The automorphism groups for the groups of orders $8p$ and $8p^2$ are given. The calculations were done using the programming language CAYLEY. Explicit presentations for both the groups of these orders and their automorphism groups are given.

Abelian Groups, Homomorphisms and Central Automorphisms of Nilpotent Groups

In this paper we find a necessary and sufficient condition for a finite nilpotent group to have an abelian central automorphism group.

The quadratic isoperimetric inequality for mapping tori of free group automorphisms I: Positive automorphisms

If $F$ is a finitely generated free group and $ϕ$ is a positive automorphism of $F$ then $F\rtimes_ϕZ$ satisfies a quadratic isoperimetric inequality.

The automorphism tower of a free group

We prove that the automorphism group of an arbitrary non-abelian free group is complete. It generalizes the result by J.Dyer and E.Formanek (1975) stating the completeness of automorphism group of finitely generated free groups. Using the description of involutions in automorphism groups of free groups (J. Dyer, P. Scott, 1975) we obtain a group-theoretic characterization of inner automorphisms determined by primitive elements in the automorphism group of any non-abelian free group F. It follows that the subgroup Inn(F) is characteristic in Aut(F), and hence the latter one is complete.

On class preserving automorphisms of groups of order 32

We study class preserving automorphisms of groups of order thirty two and prove that only two groups have non-inner class preserving automorphisms.

Subgroups and quotient groups of automorphism groups of RAAGs

We study subgroups and quotients of outer automorphism groups of right-angled Artin groups (RAAGs). We prove that for all RAAGS, the outer automorphism group is residually finite and, for a large class of RAAGs, it satisfies the Tits alternative. We also investigate which of these automorphism groups contain non-abelian solvable subgroups.

Automorphism groups of covering groups

We derive a formula connecting the orders of the automorphism groups of a finite group and of its covering groups.

关于自由群的外自同构群

On the outer automorphism groups of free groups

该研究证明了秩可数无穷的自由群的外自同构群具有完备性，丰富了自由群外自同构群相关的代数理论结论。

The Automorphism Groups of the Groups of Order 32p

The results of computer computations determining the automorphism groups of the groups of order 32$p$ for $p \geq 3$ are given in several tables. Presentations for the automorphism groups of the groups of order 32, which in many cases appear as direct product factors in the automorphism groups of order $32p$, are also presented for completeness. Many of the groups of order 32$p$ with a normal sylow $p$-subgroup have automorphism groups of the form: Hol($C_p$)$ \times $Invariant Factor. A suggestion is made as to how one might determine this invariant factor using only information on the automo…

Automorphism groups of groups of order $p^{2}q^{2}$

In this paper, we have computed the automorphism groups of all groups of order $p^{2}q^{2}$, where $p$ and $q$ are distinct primes.

The automorphism group of certain 3-generator $p$-groups

A presentation as well as a structural description of the automorphism group of a family of 3-generator finite $p$-groups is given, $p$ being an odd prime.

Contraction groups for tidy automorphisms of totally disconnected groups

We show that recent results of U. Baumgartner and G.A. Willis concerning contraction groups of automorphisms of metrizable, totally disconnected, locally compact groups remain valid also in the non-metrizable case, if one restricts attention to automorphisms for which small tidy subgroups exist.

On the Groups and Automorphism Groups of the Groups of order 64p without a Normal Sylow p-Subgroup

The groups of order 64p without a normal sylow p-subgroup are listed, and their automorphism groups are also determined. As a by-product of our original effort to get these groups, we needed to determine the automorphism groups of those groups of order 64 with an odd-order automorphism. In view of the fact that we already had determined these groups and that these automorphism groups are not given explicitly in the literature, we have appended to this report these automorphism groups. In another project we were looking for new complete groups by following automorphism group towers up to comple…

The Bass-Jiang group for automorphism-induced HNN-extensions

We use the Bass--Jiang group for automorphism-induced HNN-extensions to build a framework for the construction of tractable groups with pathological outer automorphism groups. We apply this framework to a strong form of a question of Bumagin--Wise on the outer automorphism groups of finitely presented, residually finite groups.

Palindromic Automorphisms of Free Nilpotent Groups

In this paper, we initiate the study of palindromic automorphisms of groups that are free in some variety. More specifically, we define palindromic automorphisms of free nilpotent groups and show that the set of such automorphisms is a group. We find a generating set for the group of palindromic automorphisms of free nilpotent groups of step 2 and 3. In particular, we obtain a generating set for the group of central palindromic automorphisms of these groups. In the end, we determine central palindromic automorphisms of free nilpotent groups of step 3 which satisfy the necessary condition of Br…

有限群的约化幂（有向）图的自同构群

Automorphism group of the reduced power (di)graph of a finite group

本文描述并计算了有限群的有向与无向约化幂图的完整自同构群，并建立了其与幂图自同构群之间的关系。

Growth and order of automorphisms of free groups and free Burnside groups

We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.

A Survey on Automorphism Groups of Finite $p$-Groups

This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presented with varying levels of detail, with some of the examples given in complete detail.

Finite Groups with Abelian Automorphism Groups: A Survey

In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.

A description of automorphism group of power graphs of finite groups

The power graph of a group is the graph whose vertex set is the set of nontrivial elements of group, two elements being adjacent if one is a power of the other. We introduce some way for find the automorphism groups of some graphs. As an application We describe the full automorphism group of the power graph of all finite groups. Also we obtain the full automorphism group of power graph of abelian, homocyclic and nilpotent groups

On linearity of automorphism groups of algebras and groups

The main purpose of this paper is to describe some published results and outline corresponding approaches which when applied to automorphism groups of algebras or groups establish that these groups are linear or non-linear.

Almost regular involutory automorphisms of uniquely 2-divisible groups

We prove that a uniquely 2-divisible group that admits an almost regular involutory automorphism is solvable.

A finitely presented subgroup of the automorphism group of a right-angled Artin group

Let G be a right-angled Artin group. We use geometric methods to compute a presentation of the subgroup H of Aut(G) consisting of the automorphisms that send each generator to a conjugate of itself. This generalizes a result of McCool on basis-conjugating automorphisms of free groups.

On the Automorphism Tower of Free Nilpotent Groups

In this thesis I study the automorphism tower of free nilpotent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. This allows me to lift the description of the derivation tower of the free nilpotent Lie algebra to obtain information about the automorphism tower of the free nilpotent group. The main result in this thesis states that the automorphism tower of the free nilpotent group $Γ(n,d)$ on $n$ …

Directions of automorphisms of Lie groups over local fields compared to the directions of Lie algebra automorphisms

To each totally disconnected, locally compact topological group G and each group A of automorphisms of G, a pseudo-metric space of ``directions'' has been associated by U. Baumgartner and the second author. Given a Lie group G over a local field, it is a natural idea to try to define a map from the space of directions of analytic automorphisms of G to the space of directions of automorphisms of the Lie algebra L(G) of G, which takes the direction of an analytic automorphism of G to the direction of the associated Lie algebra automorphism. We show that, in general, this map is not well-defined.…

具有循环边群的群图的自同构

Automorphisms of graphs of groups with cyclic edge groups

本文刻画了边群为循环群、顶点群无挠且中心化子循环的单端群图基本群的外自同构群，证明其本质上由顶点群外自同构、广义鲍姆斯拉格-索利塔尔群外自同构及广义扭转共同构成。

The full automorphism group of the power (di)graph of a finite group

We describe the full automorphism group of the power (di)graph of a finite group. As an application, we solve a conjecture proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.

On the automorphism groups of Frobenius Groups

This is one of a series papers which aim towards to solve the problem of determining automorphism groups of Frobenius groups. This one solves the problem in the case where the Frobenius kernels are elementary abelian and Frobenius complements are cyclic.

First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group

We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization $V$ of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism for the rank of the free group is greater than 4.

Contraction groups and scales of automorphisms of totally disconnected locally compact groups

We study contraction groups for automorphisms of totally disconnected locally compcat groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.

Automorphisms of groups and converse of Schur's theorem

An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. As consequences, we give surprisingly very easy proofs of converse of Schur's theorem and also prove some other related results.

自由幂零群的完全外自同构群

Complete outer automorphism groups of free nilpotent groups

本研究证明了2类无限生成自由幂零群的外自同构群具有完全性，推进了无限生成自由幂零群外自同构群的结构性质研究。

On the Automorphisms of Some One-Relator Groups

The description of the automorphism group of group $<a, b; [a^m,b^n]=1>$ ($m,n>1$) in terms of generators and defining relations is given. This result is applied to prove that any normal automorphism of every such group is inner.

关于acylindrically hyperbolic群的自同构、拟同构和粗自同构

On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups

研究acylindrically hyperbolic群G的自同构群对其同调拟同构空间的作用，确定其核为强共轭化自同构子群，并证明若G无非平凡有限正规子群，则其有足够多的拟同构以识别自同构是否内蕴。

Automorphisms of geometric structures associated to Coxeter groups

In this paper, we consider the automorphism groups of the Cayley graph with respect to the Coxeter generators and the Davis complex of an arbitrary Coxeter group. We determine for which Coxeter groups these automorphism groups are discrete. In the case where they are discrete, we express them as semidirect products of two obvious families of automorphisms. This extends a result of Haglund and Paulin.

Minimal odd order automorphism groups

We show that 3^7 is the smallest order of a non-trivial odd order group which occurs as the full automorphism group of a finite group.

Automorphism groups of Beauville surfaces

This paper shows that the automorphism group of a Beauville surface is a finite solvable group, and describes its possible structure. It relies on results of Singerman on triangle group inclusions, and of Lucchini on generators for special linear groups.

All automorphisms of the universal sofic group are class-preserving

In this article we prove that every automorphism of a universal sofic group is class preserving, i.e. preserves the conjugacy classes

Automorphism groups of relatively free groups of infinite rank: a survey

The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on.

Coxeter群及其自同构群的窄正规子群

Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups

本文研究Coxeter群及其自同构群中的正规子群，刻画了其中有限子群与不含非阿贝尔自由群的窄子群的结构。

On the outer automorphism groups of finitely generated, residually finite groups

Bumagin-Wise posed the question of whether every countable group can be realised as the outer automorphism group of a finitely generated, residually finite group. We give a partial answer to this problem for recursively presentable groups.

The Bieri-Neumann-Strebel Invariant of the Pure Symmetric Automorphisms of a Right-Angled Artin Group

We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.