Soluble Groups with few orbits under automorphisms
Raimundo Bastos, Alex Carrazedo Dantas, Emerson de Melo
备注：
Submitted to an international journal Geometriae Dedicata (2020)
发表信息：
Geometriae Dedicata volume 209, pages119. -- 123 (2020) - The final publication is available at https://link.springer.com/article/10.1007/s10711-020-00525-7
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 $ω(G)=3$.
It is proved that if a finite $p$-soluble group $G$ admits an automorphism $φ$ of order $p^n$ having at most $m$ fixed points on every $φ$-invariant elementary abelian $p'$-section of $G$, then the $p$-length of $G$ is bounded above in terms of $p^n$ and $m$; if in addition the group $G$ is soluble, then the Fitting height of $G$ is bounded above in terms of $p^n$ and $m$. It is also proved that if a finite soluble group $G$ admits an automorphism $ψ$ of order $p^aq^b$ for some primes $p,q$, then the Fitting height of $G$ is bounded above in terms of $|ψ|$ and $|C_G(ψ)|$.
Let $G$ be a finite group admitting a coprime automorphism $α$. Let $J_G(α)$ denote the set of all commutators $[x,α]$, where $x$ belongs to an $α$-invariant Sylow subgroup of $G$. We show that $[G,α]$ is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set $J_G(α)$ is soluble or nilpotent, respectively.
A graph is edge-primitive if its automorphism group acts primitively on the edge set, and 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs. In this paper, we present a classification for those edge-primitive graphs which are 2-arc-transitive and have soluble edge-stabilizers.
Local--global generation property of commutators in finite $π$-soluble groups
Università degli Studi di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；University of Southern California（南加州大学）；University of Lincoln（林肯大学）；University of Brasilia（巴西利亚大学）
In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.
Suppose that a finite group $G$ admits a soluble group of coprime automorphisms $A$. We prove that if, for some positive integer $m$, every element of the centralizer $C_G(A )$ has a left Engel sink of cardinality at most $m$ (or a right Engel sink of cardinality at most $m$), then $G$ has a subgroup of $(|A|,m)$-bounded index which has Fitting height at most $2α(A)+2$, where $α(A)$ is the composition length of $A$. We also prove that if, for some positive integer $r$, every element of the centralizer $C_G(A )$ has a left Engel sink of rank at most $r$ (or a right Engel sink of rank at most $r…
It is known that a finite group with an automorphism $φ$ of coprime order has a soluble radical of $(|φ|,|C_G(φ)|)$-bounded Fitting height and index. We extend this classic result as follows. Let $f(x) = a_0 + a_1 \cdot x + \cdots + a_d \cdot x^d \in \mathbb{Z}[x]$ be a primitive polynomial and let $G$ be a finite group with an automorphism $φ$ of coprime order satisfying $ g^{a_0} \cdot φ(g)^{a_1} \cdots φ^d(g)^{a_d} = 1 $, for all $g \in G$. Then the soluble radical of $G$ has $(d,|C_G(φ)|)$-boundex Fitting height and index. The bounds are made explicit and are particularly good for small va…
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…
Among other things, we prove that the group of automorphisms fixing every normal subgroup of a nilpotent-by-abelian group is nilpotent-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.
We describe under a variety of conditions abelian subgroups of the automorphism group A of the regular n-ary tree T which are normalized by the n-ary adding machine t=(e,...,e,t)s where s is the n-cycle (0,1,...,n-1). As an application, for n a prime number, and for n = 4 we prove that every soluble subgroup of A containing t is an extension of a torsion-free metabelian group by a finite group.
Let $G$ be a finite group admitting a coprime automorphism $α$ of order $e$. Denote by $I_G(α)$ the set of commutators $g^{-1}g^α$, where $g\in G$, and by $[G,α]$ the subgroup generated by $I_G(α)$. We study the impact of $I_G(α)$ on the structure of $[G,α]$. Suppose that each subgroup generated by a subset of $I_G(α)$ can be generated by at most $r$ elements. We show that the rank of $[G,α]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(α)$ has odd order, then $[G,α]$ has odd order too. Further, i…
We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_\circ^\circ$-subgroup.
For each element $u$ in a finite group $G$ define a map $θ_u\colon G\to G$ by $θ_u(g)=[g^{-u},g]$ and set $Θ_G(u)=\{g\in G\mid θ_u^n(g)=g \hbox{ for some } n>0\}$. Then $θ_u$ induces a permutation of $Θ_G(u)$; let $β_G(u)$ be the number of orbits apart from $\{1\}$. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group $G$ is bounded in terms of the values of $β_G(u)$ for $2$-elements $u$.
Suppose that a finite group $G$ admits an automorphism $φ$ of order $2^n$ such that the fixed-point subgroup $C_G(φ^{2^{n-1}})$ of the involution $φ^{2^{n-1}}$ is nilpotent of class $c$. Let $m=|C_G(φ)|$ be the number of fixed points of $φ$. It is proved that $G$ has a characteristic soluble subgroup of derived length bounded in terms of $n,c$ whose index is bounded in terms of $m,n,c$. A similar result is also proved for Lie rings.
Let $G$ be a finite group admitting a coprime automorphism $ϕ$ of order $n$. Denote by $G_ϕ$ the centralizer of $ϕ$ in $G$ and by $G_{-ϕ}$ the set $\{ x^{-1}x^ϕ; \ x\in G\}$. We prove the following results. 1. If every element from $G_ϕ\cup G_{-ϕ}$ is contained in a $ϕ$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded. 2. Suppose that $G_ϕ$ is nilpotent of class $c$. If $x^{e}=1$ for each $x \in G_{-ϕ}$ and any two elements of $G_{-ϕ}$ are contained in a $ϕ$-invariant soluble subgroup of derived length $d$, then the exponent of $[G,ϕ]$ is bounded in term…
Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is proved that $G$ has a nilpotent characteristic subgroup of index bounded in terms of $c$, $|C_G(F)|$, and $|FH|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of $C_G(F)=1$ the whole group was proved to be nilpotent of $(c,|H|)$-bounded class. Examples show that…
Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $σ(G)=\{σ_{i} : σ_{i}\cap π(G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i$ such that $σ_{i}\in σ(G)$. We say that $G$ is $σ$-full if $G$ possesses a complete Hall $σ$-set. A complete Hall $σ$-set $\cal H$ of $G$ is said to be a $σ$-basis of $G$ if every two subgroups $A, B \…
The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where the $F^*_i(G)$ is the generalized Fitting series: $F^*_1(G)=F^*(G)$ and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if $G$ admits a soluble group of automorphisms $A$ of coprime order, then $h^*(G)$ is bounded in terms of $h^* (C_G(A))$, where $C_G(A)$ is the fixed-point subgroup, and the number of prime factors of $|A|$ counting multiplicities. The result follows from the special case when $A=\langleφ\rangle$ is of prime order, where it is proved …
Let $f(x)$ be a non-zero polynomial with integer coefficients. An automorphism $φ$ of a group $G$ is said to satisfy the elementary abelian identity $f(x)$ if the linear transformation induced by $φ$ on every characteristic elementary abelian section $S$ of $G$ is annihilated by $f(x)$. We prove that if a finite (soluble) group $G$ admits a fixed-point-free automorphism $φ$ satisfying an elementary abelian identity $f(x)$, where $f(x)$ is a primitive polynomial, then the Fitting height of $G$ is bounded in terms of $\operatorname{deg}(f(x))$. We also prove that if $f(x)$ is any non-zero polyno…
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 …
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…
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…
Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $φ$ with finite Reidemeister number $R(φ)$ (the number of $φ$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite (in other words, any residually finite group of finite upper rank, which is not soluble-by-finite, has the $R_\infty$ property). This reduction is the first step in the proof of the second main theorem of the paper: suppose, $G$ is a residually finite group of finite Prüfer rank and $φ$ is its automorphism with $R(φ)<\infty$; then $R(φ)$ is equal to the number of equivalence…
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 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 $G/N \simeq A_5$.
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.
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…
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…
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.
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…
A finite non-regular primitive permutation group $G$ is extremely primitive if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, finding various applications. The classification of extremely primitive groups was recently completed by Burness and Lee, who relied on an earlier classification of soluble extremely primitive groups by Mann, Praeger and Seress. Unfortunately, there is an inaccuracy in the latter classification. We correct this mistake, and also investigate regular linear spaces which admit groups of automorphism…
Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $Γ(G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on $Γ(G,A)$.
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）
We study in detail the profinite group G arising as geometric étale iterated monodromy group of an arbitrary quadratic morphism f with an infinite postcritical orbit over a field of characteristic different from two. This is a self-similar closed subgroup of the group of automorphisms of a regular rooted binary tree. In many cases it is equal to the automorphism group of the tree, but there remain some interesting cases where it is not. In these cases we prove that the conjugacy class of G depends only on the combinatorial type of the postcritical orbit of f. We also determine the Hausdorff di…
We prove a new version of the classical peak-reduction theorem for automorphisms of free groups in the setting of right-angled Artin groups. We use this peak-reduction theorem to prove two important corollaries about the action of the automorphism group of a right-angled Artin group $A_Γ$ on the set of $k$-tuples of conjugacy classes from $A_Γ$: orbit membership is decidable, and stabilizers are finitely presentable. Further, we explain procedures for checking orbit membership and building presentations of stabilizers. This improves on a previous result of the author's. We overcome a technical…
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).
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.
