scholarly journals AUTOMORPHISMS OF THE CATEGORY OF THE FREE NILPOTENT GROUPS OF THE FIXED CLASS OF NILPOTENCY

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1273-1281 ◽  
Author(s):  
A. TSURKOV
Keyword(s):  
Algebraic Geometry ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Free Algebras ◽  
Finitely Generated ◽  
Identity Automorphism ◽  
Universal Algebraic Geometry ◽  
Inner Automorphisms

This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The key problem is to study how far the group Aut Θ0 of all automorphisms of the category Θ0 is from the group Inn Θ0 of inner automorphisms of Θ0 (see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎. Let Θ = 𝔑d be the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category [Formula: see text], d ≥ 2 is inner.

AUTOMORPHIC EQUIVALENCE OF ALGEBRAS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1263-1271 ◽  
Author(s):  
A. TSURKOV
Keyword(s):  
Algebraic Geometry ◽  
Free Algebras ◽  
Finitely Generated ◽  
Inner Automorphism ◽  
Universal Algebraic Geometry ◽  
Automorphic Equivalence

This paper is adjacent to [5] and motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The notions of geometrically equivalent algebras and weakly geometrically equivalent algebras (cf. [2]) are intimately related to the above problem (the latter notion being related to automorphisms of the category Θ0 is replaced in the present paper by a more natural one: "automorphic equivalence"). Two algebras are said to have the same algebraic geometry if they are automorphically equivalent. If an automorphic equivalence of algebras is provided by an inner automorphism of Θ0, then this equivalence coincides with the geometric equivalence. However, not all automorphisms of Θ0 are inner (see [5, Theorem 2]). In this paper we study the situation when the automorphic equivalence of algebras is provided by a non-inner automorphism. Assuming Θ to be an IBM variety of one-sorted algebras we prove that two algebras H1 and H2 are automorphically equivalent in Θ if and only if H1 is geometrically equivalent to [Formula: see text], where [Formula: see text] is the algebra with the same domain as H2 and with the new operations induced by some system of words W = {wω(x)|ω ∈ Ω, Ω is the signature of algebras in Θ. Thus, the notion of automorphic equivalence is reduced to a more simple one of geometric equivalence.

scholarly journals Some properties of the growth and of the algebraic entropy of group endomorphisms

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Pablo Spiga
Keyword(s):  
Finite Groups ◽  
Addition Theorem ◽  
Finitely Generated ◽  
Growth Type ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Algebraic Entropy ◽  
Finitely Generated Groups ◽  
Identity Automorphism ◽  
Inner Automorphisms

AbstractWe study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.

IA-automorphisms of finitely generated nilpotent groups

2014 ◽  
Vol 13 (07) ◽  
pp. 1450027 ◽  
Author(s):  
Sandeep Singh ◽  
Deepak Gumber ◽  
Hemant Kalra
Keyword(s):  
Nilpotent Groups ◽  
Finitely Generated ◽  
Identity Automorphism

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) ≃ Inn (G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner.

scholarly journals Separability properties of automorphisms of graph products of groups

2016 ◽  
Vol 26 (01) ◽  
pp. 1-27
Author(s):  
Michal Ferov
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Technical Condition ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Residually Finite ◽  
Products Of Groups ◽  
Text Version ◽  
Inner Automorphisms

We study properties of automorphisms of graph products of groups. We show that graph product [Formula: see text] has nontrivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has nontrivial pointwise inner automorphisms. We use this result to study residual finiteness of [Formula: see text]. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then [Formula: see text] is residually finite. Finally, we generalize this result to graph products of residually [Formula: see text]-finite groups to show that if [Formula: see text] is a graph product of finitely generated residually [Formula: see text]-finite groups such that the vertex groups corresponding to central vertices satisfy the [Formula: see text]-version of the technical condition then [Formula: see text] is virtually residually [Formula: see text]-finite. We use this result to prove bi-orderability of Torreli groups of some graph products of finitely generated residually torsion-free nilpotent groups.

Equality of derival automorphisms and certain automorphisms in finitely generated groups

2019 ◽  
Vol 18 (05) ◽  
pp. 1950088
Author(s):  
Zahedeh Azhdari
Keyword(s):  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Inner Automorphisms ◽  
Central Automorphisms

Let [Formula: see text] be a group and [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote the group of all inner automorphisms, the group of all pointwise inner automorphisms, the group of all central automorphisms and the group of all derival automorphisms of [Formula: see text], respectively. We know that in a finite [Formula: see text]-group [Formula: see text] of class 2, [Formula: see text] if and only if [Formula: see text] is cyclic and [Formula: see text], where [Formula: see text] is the group of all derival automorphisms of [Formula: see text] which fix [Formula: see text] elementwise. In this paper, we characterize all finite nilpotent groups of class 2 for which [Formula: see text] or [Formula: see text] is equal to [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we characterize all finitely generated nilpotent groups of class 2 for which [Formula: see text] is equal to [Formula: see text] and give some interesting corollaries in this regard.

scholarly journals COMBING NILPOTENT AND POLYCYCLIC GROUPS

1999 ◽  
Vol 09 (02) ◽  
pp. 135-155 ◽  
Author(s):  
ROBERT H. GILMAN ◽  
DEREK F. HOLT ◽  
SARAH REES
Keyword(s):  
Real Time ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Heisenberg Groups ◽  
Finitely Generated ◽  
Language Class ◽  
Automatic Groups ◽  
The Family ◽  
Polycyclic Groups ◽  
Cyclic Commutator

The notable exclusions from the family of automatic groups are those nilpotent groups which are not virtually abelian, and the fundamental groups of compact 3-manifolds based on the Nil or Sol geometries. Of these, the 3-manifold groups have been shown by Bridson and Gilman to lie in a family of groups defined by conditions slightly more general than those of the automatic groups, i.e. to have combings which lie in the formal language class of indexed languages. In fact, the combings constructed by Bridson and Gilman for these groups can also be seen to be real-time languages (i.e. recognized by real-time Turing machines). This article investigates the situation for nilpotent and polycyclic groups. It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are all 2 or 3-generated class 2 nilpotent groups, and groups in specific families of nilpotent groups (the finitely generated Heisenberg groups, groups of unipotent matrices over Z and the free class 2 nilpotent groups). Further, it is shown that any polycyclic-by-finite group embeds in a real-time combable group. All the combings constructed in the article are boundedly asynchronous, and those for nilpotent-by-finite groups have polynomially bounded length functions, of a degree equal to the nilpotency class, c; this verifies a polynomial upper bound on the Dehn functions of those groups of degree c+1.

scholarly journals Automorphic equivalence of linear algebras

2014 ◽  
Vol 13 (07) ◽  
pp. 1450026 ◽  
Author(s):  
A. Tsurkov
Keyword(s):  
Algebraic Geometry ◽  
Infinite Field ◽  
Associative Algebras ◽  
Closed Sets ◽  
First Case ◽  
Invertible Elements ◽  
Universal Algebraic Geometry ◽  
Inner Automorphisms ◽  
Automorphic Equivalence ◽  
First Time

This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras Θ and algebras H ∈ Θ from this variety. One of the central question of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras H1 and H2 are called geometrically equivalent if and only if the H1-closed sets coincide with the H2-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras H1 and H2 are called automorphically equivalent if and only if the H1-closed sets coincide with the H2-closed sets after some "changing of coordinates". We can detect the difference between geometric and automorphic equivalence of algebras of the variety Θ by researching of the automorphisms of the category Θ0 of the finitely generated free algebras of the variety Θ. By [5] the automorphic equivalence of algebras provided by inner automorphism coincide with the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety Θ if the factor group 𝔄/𝔜 is big. Here 𝔄 is the group of all automorphisms of the category Θ0, 𝔜 is a normal subgroup of all inner automorphisms of the category Θ0. In [6] the variety of all Lie algebras and the variety of all associative algebras over the infinite field k were studied. If the field k has not nontrivial automorphisms then group 𝔄/𝔜 in the first case is trivial and in the second case has order 2. We consider in this paper the variety of all linear algebras over the infinite field k. We prove that group 𝔄/𝔜 is isomorphic to the group (U(kS2)/U(k{e}))λ Aut k, where S2 is the symmetric group of the set which has 2 elements, U(kS2) is the group of all invertible elements of the group algebra kS2, e ∈ S2, U(k{e}) is a group of all invertible elements of the subalgebra k{e}, Aut k is the group of all automorphisms of the field k. So even the field k has not nontrivial automorphisms the group 𝔄/𝔜 is infinite. This kind of result is obtained for the first time. The example of two linear algebras which are automorphically equivalent but not geometrically equivalent is presented in the last section of this paper. This kind of example is also obtained for the first time.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Powerfully nilpotent groups of rank 2 or small order

2020 ◽  
Vol 23 (4) ◽  
pp. 641-658
Author(s):  
Gunnar Traustason ◽  
James Williams
Keyword(s):  
Detailed Analysis ◽  
Special Kind ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Rank 2 ◽  
Full Classification

AbstractIn this paper, we continue the study of powerfully nilpotent groups. These are powerful p-groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank 2. The classification will then be used to arrive at a precise formula for the number of powerfully nilpotent groups of rank 2 and order {p^{n}}. We will also give a detailed analysis of the ancestry tree for these groups. The second part of the paper is then devoted to a full classification of powerfully nilpotent groups of order up to {p^{6}}.

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

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