scholarly journals A sufficient condition for finite presentability of abelian-by-nilpotent groups

2014 ◽  
Vol 8 (1) ◽  
pp. 257-283
Author(s):  
Ralph Strebel
Keyword(s):  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Presentability

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

scholarly journals Groups, nonadditive entropy and phase transitions

2014 ◽  
Vol 28 (24) ◽  
pp. 1450162 ◽  
Author(s):  
Nikos Kalogeropoulos
Keyword(s):  
Growth Rate ◽  
Phase Transitions ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Tsallis Entropy ◽  
Discrete Groups ◽  
Sufficient Condition ◽  
Thermodynamic Behavior ◽  
Nonadditive Entropy ◽  
Random Groups

We investigate the possibility of discrete groups furnishing a kinematic framework for systems whose thermodynamic behavior may be given by nonadditive entropies. Relying on the well-known result of the growth rate of balls of nilpotent groups, we see that maintaining extensivity of the entropy of a nilpotent group requires using a non-Boltzmann/Gibbs/Shannon (BGS) entropic form. We use the Tsallis entropy as an indicative alternative. Using basic results from hyperbolic and random groups, we investigate the genericity and possible range of applicability of the BGS entropy in this context. We propose a sufficient condition for phase transitions, in the context of (multi-) parameter families of nonadditive entropies.

Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups

1999 ◽  
Vol 42 (3) ◽  
pp. 335-343 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Residual Finiteness ◽  
Necessary And Sufficient Condition ◽  
Residual Properties ◽  
Hnn Extensions ◽  
Subgroup Separability ◽  
Necessary And Sufficient

AbstractWe derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties ofHNN-extensions of nilpotent groups with cyclic associated subgroups.

scholarly journals On finitary properties for fiber products of free semigroups and free monoids

2020 ◽  
Vol 101 (2) ◽  
pp. 326-357
Author(s):  
Ashley Clayton
Keyword(s):  
Sufficient Conditions ◽  
Free Monoid ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Free Monoids ◽  
Fiber Product ◽  
Finite Presentability ◽  
Free Semigroups ◽  
Necessary And Sufficient

Abstract We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and $$\mathcal {J}$$ J -trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.

The Current Situation in Chemical Stabilization of Biological Materials

1972 ◽  
Vol 30 ◽  
pp. 132-133
Author(s):  
John H. Luft
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Osmium Tetroxide ◽  
Molecular Level ◽  
Cross Linking ◽  
Chemical Stabilization ◽  
Specimen Preparation ◽  
Sufficient Condition ◽  
Radio Telescopes

With information processing devices such as radio telescopes, microscopes or hi-fi systems, the quality of the output often is limited by distortion or noise introduced at the input stage of the device. This analogy can be extended usefully to specimen preparation for the electron microscope; fixation, which initiates the processing sequence, is the single most important step and, unfortunately, is the least well understood. Although there is an abundance of fixation mixtures recommended in the light microscopy literature, osmium tetroxide and glutaraldehyde are favored for electron microscopy. These fixatives react vigorously with proteins at the molecular level. There is clear evidence for the cross-linking of proteins both by osmium tetroxide and glutaraldehyde and cross-linking may be a necessary if not sufficient condition to define fixatives as a class.

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.

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