G. Endimioni
An automorphism of a group is said to be normal if it preserves each normal subgroup. In this paper, we determine the normal automorphisms of a free metabelian nilpotent group.
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We construct an example of an IA-automorphism of the free metabelian group of rank $n\geq 3$ without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain. By a result of Bachmuth \cite{Ba1}, such an automorphism does not exist if the rank is equal to 2.
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.
Automorphisms of a Free Centre-by-Centre-by-Metabelian Group of Rank 3
研究3阶自由中心-中心-可解群的自同构群，发现其包含一个稠密于形式幂级数拓扑的有限生成子群。
We prove that every free metabelian non--cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary we prove that for every prime number $p$ an arbitrary free metabelian non--cyclic group has a finitely generated $p'$--isolated subgroup which is not $p$--separable.
Regular maps from the lamplighter to metabelian groups
本文证明灯夫群可单射利普希茨映射到任何非虚拟幂零的有限生成亚阿贝尔群，从而说明这类群满足Hume等人提出的解析薄/解析厚二分法。
Automorphisms of Free Metabelian Lie Algebras, II
该研究证明了秩n≥4的自由亚阿贝尔李代数Mₙ的自同构几乎可驯服，且特征≠3时Aut(Mₙ)由线性、二次、三次自同构生成，n≥5时任意特征域均成立。
For a positive integer $n$, with $n \geq 2$, let $M_n$ be a free metabelian group of rank $n$. For $c \in \mathbb{N}$, let $γ_c(M_n)$ be the $c$-th term of the lower central series of $M_n$. For $c \geq 2$, let ${\rm I}_{c}{\rm A}(M_n)$ be the subgroup of ${\rm Aut}(M_{n})$ consisting of all automorphisms inducing the identity mapping on $M_n/γ_c(M_n)$. In this paper, we study the quotient groups ${\cal L}^{c}({\rm IA}(M_{n})) = {\rm I}_{c}{\rm A}(M_n)/{\rm I}_{c+1}{\rm A}(M_n)$ for all $n$ and $c$. For $c \geq 2$, we show $γ_{c}({\rm IA}(M_{2})) = {\rm I}_{c+1}{\rm A}(M_{2}))$. For $n = 3$, w…
On the critical regularity of nilpotent groups acting on the interval: the metabelian case
本文研究无挠有限生成幂零亚阿贝尔群在区间上保定向微分同胚群中的嵌入正则性，证明临界正则性由商群无挠秩决定，并以海森堡群为例给出精确临界值。
Juhasz has proved that the automorphism group of a group G of maximal class of order p^n, with p\ge 5 and n>p+1, has order divisible by $p^{\lceil(3n-2p+5)/2\rceil}$. We show that by translating the problem in terms of derivations, the result can be deduced from the case where G is metabelian. Here one can use a general result of Caranti and Scoppola concerning automorphisms of two-generator, nilpotent metabelian groups.
Automorphisms of free metabelian Lie algebras, I
本文研究了自由双交换李代数的自同构性质，证明了特定条件下自同构的 tame 性及部分自同构的几乎 tame 性，推翻了领域内长期存在的结论。
We prove that if a group is nilpotent (resp. metabelian), then so is the subgroup of its automorphism group generated by all polynomial automorphisms.
In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.
In this paper we find a necessary and sufficient condition for a finite nilpotent group to have an abelian central automorphism group.
A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Bieri and R. Strebel in 1980.
We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "…
We describe the nilpotent subgroups of the group Bir(P^2(C)) of birational transformations of the complex projective plane. Let N be a nilpotent subgroup of class k>1; then either each element of N has finite order, or N is virtually metabelian.
In this paper, we present an explicit structure for the Baer invariant of a free nilpotent group (the $n$-th nilpotent product of the infinite cyclic group, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*... \st{n}*\textbf{Z}$) with respect to the variety of polynilpotent groups of class row $(c,1)$, ${\cal N}_{c,1}$, for all $c > 2n-2$. In particular, an explicit structure of the Baer invariant of a free abelian group with respect to the variety of metabelian groups will be presented.
Strong conciseness in profinite metabelian groups
该研究证明每个群字在 profinite 亚阿贝尔群类中具有强简洁性，并借助新方法给出了 profinite 幂零群类中对应结论的另一种证明。
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。
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 $n$ …
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…
A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the metabelian small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also prove a necessary condition for the capability of an arbitrary p-group of class k, and some further results.
Dyer and Formanek (1976) proved that if N is a free nilpotent group of class two and of finite rank which is not equal to 1, or to 3, then the automorphism group Aut(N) of N is complete. The main result of the present paper states that the automorphism group of any infinitely generated free nilpotent group of class two is also complete.
We investigate some situation in which automorphisms of a group G are uniquely determined by their restrictions to a proper subgroup H. Much of the paper is devoted to studying under which additional hypotheses this property forces G to be nilpotent if H is. As an application we prove that certain countably infinite locally nilpotent groups have uncountably many (outer) automorphisms.
In arXiv:1303.1129, the authors provided a bound for the palindromic width of free abelian-by-nilpotent group $AN_n$ of rank $n$ and free nilpotent group ${\rm N}_{n,r}$ of rank $n$ and step $r$. In the present paper we study palindromic widths of groups $\widetilde{AN}_n$ and $\widetilde{\rm N}_{n,r}$. We denote by $\widetilde{G}_n = G_n / \langle \langle x_1^2, \ldots, x_n^2 \rangle \rangle$ the quotient of group $G_n = \langle x_1, \ldots, x_n \rangle$, which is free in some variety by the normal subgroup generated by $x_1^2, \ldots, x_n^2$. We prove that the palindromic width of the quotie…
For a positive integer $n$, with $n \geq 4$, let $R_{n}$ be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank $n$. We show that the subgroup of Aut$(R_{n})$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_{n}$ is dense in Aut$(R_{n})$ with respect to the formal power series topology.
We exhibit normal subgroups of a free nilpotent group F of rank two and class three, which have isomorphic finite quotients but are not conjugate under any automorphism of F.
We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely $H_3(\mathbb{Z})$ the three-dimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann.
Assume $G$ is a finite $p$-group, and let $S$ be a Sylow $p$-subgroup of $\operatorname{Aut}(G)$ with $\exp(S)=q$. We prove that if $G$ is of class $c$, then $\exp(G)|p^{\ceil{\log_pc}}q^3$, and if $G$ is a metabelian $p$-group of class at most $2p-1$, then $\exp(G)|pq^3$.
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…
Let $G$ be a connected, simply connected one-parameter metabelian nilpotent Lie group, that means, the corresponding Lie algebra has a one-codimensional abelian subalgebra. In this article we show that $G$ contains a discrete cocompact subgroup. Given a discrete cocompact subgroup $Γ$ of $G$, we define the quasi-regular representation $τ= {\rm ind}_Γ^G 1$ of $G$. The basic problem considered in this paper concerns the decomposition of $τ$ into irreducibles. We give an orbital description of the spectrum, the multiplicity function and we construct an explicit intertwining operator between $τ$ a…
Virtually nilpotent groups with finitely many orbits under automorphisms
本文研究几乎幂零群在自同构作用下具有有限个轨道的结构，证明了群可分解为挠子群与无挠幂零可除特征子群的半直积，并给出导子群的分解及在无挠情况下的幂零性结论。
The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [4]. Later, this result was extended to all abelian groups [3] and, recently, to all torsion finitely quasihamiltonian groups [7]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [8]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $C\left(IA(Φ_{n}),Φ_{n}\right)$, where $Φ_{n}$ is a free metabelian group on $n\geq4$ generators, and $IA(Φ_{n})=\ker(Aut(Φ_{n})\to GL_{n}(\mathbb{Z}))$. We show that in this case $C(IA(Φ_{n}),Φ_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happen…
A connected Lie group admitting an expansive automorphism is known to be nilpotent, but all nilpotent Lie groups do not admit expansive automorphism. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphism.
We describe the Reidemeister spectrum $Spec_RG$ for $G = N_{rc},$ the free nilpotent group of rank $r$ and class $c,$ in the cases: $r \in {\mathbb N}$ and $c = 1;$ $r = 2, 3$ and $c = 2;$ $ r = 2$ and $c = 3,$ and prove that any group $N_{2c}$ for $c \geq 4$ satisfies to the $R_{\infty}$ property. As a consequence we obtain that every free solvable group $S_{2t}$ of rank 2 and class $t \geq 2$ (in particular the free metabelian group $M_2 = S_{22}$ of rank 2) satisfies to the $R_{\infty}$ property. Moreover, we prove that any free solvable group $S_{rt}$ of rank $r \geq 2$ and class $t$ big e…
Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator quotient, the information content of each coclass subtree with metabelian mainline is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which…
Let $G$ be a group. The central automorphism group $Aut_c(G)$ of $G$ is the centralizer of $Inn(G)$ the subgroup of $Aut(G)$ of inner automorphisms. There is a one to one map $ σ\mapsto h_σ$ from the set $Aut_c(G)$ onto the set $Hom(G,Z(G))$ of homomorphisms from $G$ onto its center, with $ h_σ(x)=x^{-1} σ(x)$. This map can be used to obtain informations about the size of $Aut_c(G)$, and also about its structure in some special cases. In this paper we see how to use it to obtain informations about the structure of $Aut_c(G)$ in the general case. The notion of the adjoint group of a ring is the…
In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic $p$-group. Lastly, we n…
The pro-isomorphic zeta function of a finitely generated nilpotent group $Γ$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $Γ$. Such zeta functions can be expressed as Euler products of $p$-adic integrals over the $p$-adic points of an algebraic automorphism group associated to $Γ$. In this way they are closely related to classical zeta functions of algebraic groups over local fields. We describe the algebraic automorphism groups for a natural family of class-$2$ nilpotent groups; these groups can be viewed as gene…
An argument used to show that certain varieties of nilpotent groups have instances of nontrivial dominions is considered, and generalized. The same is done with the argument used to show that there are nontrivial dominions in the variety of metabelian groups, to suggest how this general technique may be used.
Some three-dimensional non-nilpotent Leibniz algebras: automorphism groups
本文研究了三维非幂零 Leibniz 代数，明确给出了两类该代数的自同构群的具体结构。
Self-similarity of some soluble relatively free groups
研究可解相对自由群的传递自相似性，证明有限秩自由幂零群具有该性质，而秩至少为2的自由亚Abel群不具有。
On some relatively free pro-p groups
本文证明了在2个生成元上的相对自由中心-by-metabelian pro-p群与（2类幂零）-by-abelian pro-p群在所有profinite群类中均有限可公理化。
Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently for metabelian groups, by Eisele and Margolis. However it is known to be true for many classes of solvable groups, as for example nilpotent groups, cyclic-by-abelian groups and groups having a cyclic Sylow subgroup with abelian complement. On the other hand, it is not known whether the conjecture holds for supersolvable groups. This paper is a contribution …
Considering a particular case of a problem posed by Saharon Shelah, we prove that the automorphism group of an infinitely generated free nilpotent group N first-order interprets the full second-order theory of the set rank(N) in the empty language.
Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. The following results are proved. If $γ_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then $γ_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. Earlier this was known only in the case where $k\leq 4$.
We study the word length entropy of automorphisms of residually nilpotent groups, and how the entropy of such group automorphisms relates to the entropy of induced automorphisms on various nilpotent quotients. We show that much like the structure of a nilpotent group is dictated to a large degree by its abelianization, the entropy of an automorphism of a nilpotent group is dictated by its entropy on the abelianization. We give some applications to the study of pseudo-Anosov homeomorphisms of surfaces. In particular, we show that if $ψ$ is a non--homological pseudo-Anosov homeomorphism of a sur…
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.
A residually finite analogue of Kegel's theorem on splitting automorphisms
本文证明，若周期剩余有限群G具有素数阶分裂自同构，则G是幂零群且幂零类由p界定，从而对剩余有限群肯定了Sozutov问题。
If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isomorphism, our ultimate concern. It turns out that many of the groups in a given equivalence class share various properties including finite presentability. There are examples, such as the lamplighter group, where an equivalence class consists of a single isomorphism class and othe…
For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.
A dessin is an embedding of connected bipartite graph into an oriented closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges. In the present paper regular dessins with a nilpotent automorphism group are investigated, and attention are paid on those with the highest level of external symmetry. Depending on the algebraic theory of dessins and using group-theoretical methods, we present a classification of nilpotent groups of class two which underly a unique regular dessin.
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.
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…
It was shown by Lubotzky in 2014 that automorphism invariant subsets of finite simple groups which contain identity are always word images. In this article, we study word maps on finite nilpotent groups and show that for arbitrary finite groups, the number of automorphism invariant subsets containing identity which are not word images, referred to as word image impostors, may be arbitrarily larger than the number of actual word images. In the course of it, we construct a $2$-exhaustive set of word maps on nilpotent groups of class $2$ and demonstrate its minimality in some cases.
Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and complement $H$, acts by automorphisms on a finite group $G$, in such a way that $C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is nilpotent and its nilpotency class can be bounded in terms of $|H|$ and the nilpotency class of $C_G(H)$. Until now, it was not clear whether the bound could be made independent of the order of $H$. In this article, we construct a family $\mathfrak{G}$ of finite nilpotent groups, of unbounded nilpotency class. Each group in $\mathfrak{G}$ admits a metacyclic Frobenius group of a…
Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free nilpotent group must be very short. Let F_{n,c} denote a free nilpotent group of finite rank n at least two and of nilpotency class c at least two. In 1976 Dyer and Formanek proved that the automorphism group of F_{n,2} is even complete (and hence the height of the aumorphism tower of F_{n,2} is two) provided that n is not three; in the case when n=3, the height of the automorphism tower of F_{n,2} is three. The author proved in 2001 that the automorphism group of any infinitely generated free nilpotent …
The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $a\in A^{\#}$. Then the group $G$ is locally nilpotent.
New nilpotent series are produced that refine the usual nilpotent series of a group. These refinements can be arbitrarily longer than the series they refine and therefore clarify in greater detail the structure of automorphisms of nilpotent groups. Examples, properties, and an application to group-isomorphism testing are provided.
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