We give a complete classification of the finite $2$-groups $G$ for which the automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has three orbits. There are two infinite families and one additional group, of order $2^9$. All of them are Suzuki $2$-groups, and they appear in an earlier classification of Dornhoff.
Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. In this paper we prove that if $G$ is an FC-group with finitely many automorphism orbits, then the derived subgroup $G'$ is finite and $G$ admits a decomposition $G = Tor(G) \times D$, where $Tor(G)$ is the torsion subgroup of $G$ and $D$ is a divisible characteristic subgroup of $Z(G)$. We also show that if $G$ is an infinite FC-group with $ω(G) \leqslant 8$, then either $G$ is soluble or $G \cong A_5 \times H$, whe…
Let $G$ be a group. The orbits of the natural action of Aut$(G)$ on $G$ are called ``automorphism orbits'' of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. We prove that if $G$ is a soluble group with finite rank such that $ω(G)< \infty$, then $G$ contains a torsion-free characteristic nilpotent subgroup $K$ such that $G = K \rtimes H$, where $H$ is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that $ω(G)=3$.
Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. In this paper the finite nonsolvable groups $G$ with $ω(G) \leq 6$ are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite nonsolvable groups $G$ with $ω(G)=7$. Moreover it is proved that for a given number $n$ there are only finitely many finite groups $G$ without nontrivial abelian normal subgroups and such that $ω(G) \leq n$, generalizing a re…
Automorphism groups of Cayley graphs on almost simple groups with normal connection sets
Institute for Mathematical Sciences, Renmin University of China（中国人民大学数学科学研究所）；Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院数学与复杂系统教育部重点实验室）；School of Mathematics and Statistics The University of Melbourne（墨尔本大学数学与统计学院）
To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function Φ_w that encodes information on the orbit structure of w. We prove that this is a rational function if w can be described by finitely many recursion relations of a particular form. We show that this condition is satisfied for all elements of the discrete iterated monodromy group Γassociated to a postcritically finite quadratic polynomial over C. For such Γwe also prove that there are only finitely many possibilities for the denominator of Φ_w, and we describe a procedure to determi…
We study finite groups $G$ such that the maximum length of an orbit of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ is bounded from above by a constant. Our main results are the following: Firstly, a finite group $G$ only admits $\operatorname{Aut}(G)$-orbits of length at most $3$ if and only if $G$ is cyclic of one of the orders $1$, $2$, $3$, $4$ or $6$, or $G$ is the Klein four group or the symmetric group of degree $3$. Secondly, there are infinitely many finite ($2$-)groups $G$ such that the maximum length of an $\operatorname{Aut}(G)$-orbit on $G$ is $8$. T…
We study the nonabelian composition factors of a finite group $G$ assumed to admit an $\operatorname{Aut}(G)$-orbit of length at least $ρ|G|$, for a given $ρ\in\left(0,1\right]$. Our main results are the following: The orders of the nonabelian composition factors of $G$ are then bounded in terms of $ρ$, and if $ρ>\frac{18}{19}$, then $G$ is solvable. On the other hand, for each nonabelian finite simple group $S$, there is a constant $c(S)\in\left(0,1\right]$ such that $S$ occurs with arbitrarily large multiplicity as a composition factor in some finite group $G$ having an $\operatorname{Aut}(G…
For $N\geq 4$, we show that there exist automorphisms of the free group $F_N$ which have a parabolic orbit in $\partial F_N$. In fact, we exhibit a technology for producing infinitely many such examples.
Let $Ω$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $Ω^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph $H(n,m)$, by using elementary facts of group theory and graph theory.
For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ a…
Denote by $ω(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $ω(G) \leqslant 5$, then $G$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. We also consider the case when $ω(G) = 6$ and show that if $G$ is a nonsolvable finite group with $ω(G) = 6$, then either $G \simeq PSL(3,4)$ or there exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.
Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism of $H$. This leads naturally to a definition of "inner automorphism" applicable to the objects of any category. Bergman and Hofstra--Parker--Scott have computed these inner automorphisms for various structures including $k$-algebras, monoids, lattices, unital rings, and quandles---showing that, in each case, they are given by an obvious notion of conjugation…
We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism groups which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree. Our results further cover several interesting finitely generated subgroups of the tree almost automorphism groups, including the Thompson groups $F$, $T$, and $V$. We show in particular that Thompson's group $T$ has no commensurated subgroups other…
It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is red…
If H is a flat group of automorphisms of finite rank n of a totally disconnected, locally compact group G, then each orbit of H in the metric space B(G) of compact, open subgroups of G is quasi-isometric to n-dimensional euclidean space. In this note we prove the following partial converse: Assume that G is a totally disconnected, locally compact group such that B(G) is a proper metric space and let H be a group of automorphisms of G such that some (equivalently every) orbit of H in B(G) is quasi-isometric to n-dimensional euclidean space, then H has a finite index subgroup which is flat of ra…
We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F_2? Moreover, decidability subsists if we allow H to be a rational subset of F_2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, we show the decidability of the following weaker problem: given a finitely generated subgroup H, a word u and an integer k, d…
This article studies the complexity of the word problem in groups of automorphisms of subshifts. We show in particular that for any Turing degree, there exists a subshift whose automorphism group contains a subgroup whose word problem has exactly this degree.
For a finite group $G$, we denote by $ω(G)$ the number of $\operatorname{Aut}(G)$-orbits on $G$, and by $\operatorname{o}(G)$ the number of distinct element orders in $G$. In this paper, we are primarily concerned with the two quantities $\mathfrak{d}(G):=ω(G)-\operatorname{o}(G)$ and $\mathfrak{q}(G):=ω(G)/\operatorname{o}(G)$, each of which may be viewed as a measure for how far $G$ is from being an AT-group in the sense of Zhang (that is, a group with $ω(G)=\operatorname{o}(G)$). We show that the index $|G:\operatorname{Rad}(G)|$ of the soluble radical $\operatorname{Rad}(G)$ of $G$ can be …
We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group $C$ can be realized as a group of outer automorphisms of a group $N$, where $N$ is a finitely generated group having Kazhdan's property (T) and containing exactly two conjugacy classes.
In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $Γ$, $Γ\not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\la…
We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $γ_2(G) < \mathrm{Z}(G) < Φ(G)$, where $γ_2(G)$, $\mathrm{Z}(G)$ and $Φ(G)$ denote the commutator subgroup, the center and the Frattini subgroup of $G$ respectively. For a finite $p$-group $G$ with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) $\mathrm{Z}(G) = Φ(G)$ is elementary abelian; (ii) $γ_2(G) = Φ(G)$ is elementar…
We present an effective algorithm for detecting automorphic orbits in free groups, as well as a number of algorithmic improvements of train tracks for free group automorphisms.
We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts ``like an automorphism" on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological properties of the corresponding one-relator group. These elements (``generalized primitive elements"), interesting in their own right, can also be used for distinguishing automorphisms among arbitrary endomo…
Blocks with a single automorphism orbit of irreducible Brauer characters
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）；Faculty of Mathematics and Statistics, Hubei University（湖北大学数学与统计学院）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and quasi-invariant measures, the induced actions on the spaces of probability measures on the groups, and results concerning various issues in ergodic theory, topological dynamics, smooth dynamical systems, and probability theory on Lie groups.
We study the conjugacy problem in the automorphism group $Aut(T)$ of a regular rooted tree $T$ and in its subgroup $FAut(T)$ of finite-state automorphisms. We show that under the contracting condition and the finiteness of what we call the orbit-signalizer, two finite-state automorphisms are conjugate in $Aut(T)$ if and only if they are conjugate in $FAut(T)$, and that this problem is decidable. We prove that both these conditions are satisfied by bounded automorphisms and establish that the (simultaneous) conjugacy problem in the group of bounded automata is decidable.
Motivated by the theory of Riemann surfaces, we classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case we also give information about the branching datum of the surface. There is a shorter version of this article (submitted for publication), and in this extended version we give many more details about the GAP code that we use for the calculations. We also explicitly include a le…
We determine the groups of automorphisms and their orbits for nilpotent Lie algebras of class 2 and small dimension, over arbitrary fields (including the characteristic 2 case).
Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for S into itself, and show that any finite index subgroup of the Torelli group for S is co-Hopfian.
The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out that any such automorphism leaves $\mathcal{O}_2$ globally invariant. Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what the group…
Study of the dynamics of automorphisms of a group is usually focused on their growth and/or finite orbits, including fixed points. In this paper, we introduce properties of a different kind; using somewhat informal language, we call them metric properties. Two principal characteristics of this kind are called here the "curl" and the "flux"; there seems to be very little correlation between these and the growth of an automorphism, which means they are likely to be an essentially new tool for studying automorphisms. We also observe that our definitions of the curl and flux are sufficiently gener…
We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of a birational action of an infinite Coxeter group on ordered point sets in the projective plane modulo projective equivalence. We also outline the classification of non-rational surfaces with large automorphism groups.
A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset $S\subset\mathbb{C}\mathbb{P}^{1}$. The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collection…
Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this result to the situation of a Krull-Schmidt category on which a finite group acts as automorphisms. This then provides the orbit category introduced by Cibils and Marcos, and studied intensively by Keller in the context of cluster algebras, and by Asashiba in the context of Galois covering functors. We formulate and prove Clifford's theorem for Krull-Schmidt orbit categories with respect to a finite…
We study the action of the group of polynomial automorphisms of C^n (n>2) which preserve the Markoff-Hurwitz polynomial H(x):= x_1^2 + x_2^2 + ... + x_n^2 - x_1 x_2 ... x_n. Our main results include the determination of the group, the description of a non-empty open subset of C^n on which the group acts properly discontinuously (domain of discontinuity), and identities for the orbit of points in the domain of discontinuity.
