On groups with action on itself

2019 ◽  
Vol 26 (3) ◽  
pp. 459-470
Author(s):  
Enver Önder Uslu ◽  
Ahmet Faruk Aslan ◽  
Alper Odabaş

Abstract We introduce the notions of center, singularity and nilpotency (class) of a group with action on itself. Also, we describe a new package GwA for GAP4, including functions checking some properties of groups with action on itself with finite underlying group. As applications of the implemented functions, we give examples of groups with action satisfying Condition 1 stated in [3]. In other words, we get concrete examples of “coquecigrue” in the terminology of Loday [8, 9].

2017 ◽  
Author(s):  
Indra Riyanto ◽  
Lestari Margatama

The recent degradation of environment quality becomes the prime cause of the recent occurrence of natural disasters. It also contributes in the increase of the area that is prone to natural disasters. Flood history data in Jakarta shows that flood occurred mainly during rainy season around January – February each year, but the flood area varies each year. This research is intended to map the flood potential area in DKI Jakarta by segmenting the Digital Elevation Model data. The data used in this research is contour data obtained from DPP–DKI with the resolution of 1 m. The data processing involved in this research is extracting the surface elevation data from the DEM, overlaying the river map of Jakarta with the elevation data. Subsequently, the data is then segmented using watershed segmentation method. The concept of watersheds is based on visualizing an image in three dimensions: two spatial coordinates versus gray levels, in which there are two specific points; that are points belonging to a regional minimum and points at which a drop of water, if placed at the location of any of those points, would fall with certainty to a single minimum. For a particular regional minimum, the set of points satisfying the latter condition is called the catchments basin or watershed of that minimum, while the points satisfying condition form more than one minima are termed divide lines or watershed lines. The objective of this segmentation is to find the watershed lines of the DEM image. The expected result of the research is the flood potential area information, especially along the Ciliwung river in DKI Jakarta.


2020 ◽  
Vol 23 (4) ◽  
pp. 641-658
Author(s):  
Gunnar Traustason ◽  
James Williams

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}}.


2017 ◽  
Vol 16 (11) ◽  
pp. 1750205
Author(s):  
Özge Öztekin ◽  
Naime Ekici

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.


1977 ◽  
Vol 17 (1) ◽  
pp. 53-89 ◽  
Author(s):  
David Shield

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.


2020 ◽  
Vol 12 (1) ◽  
pp. 199-207
Author(s):  
I.Yu. Raievska ◽  
M.Yu. Raievska

It is proved that for ${p>2}$ every finite non-metacyclic $2$-generated p-group of nilpotency class $2$ with cyclic commutator subgroup is the additive group of a local nearring and in particular of a nearring with identity. It is also shown that the subgroup of all non-invertible elements of this nearring is of index $p$ in its additive group.


2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Maryam A. Alghamdi ◽  
Sompong Dhompongsa ◽  
Naseer Shahzad

A common fixed point theorem for a pair of maps satisfying condition (C) is proved under certain conditions. We extend the well-knownDeMarr's fixed point theorem to the case of noncommuting family of maps satisfying condition (C). As for an application, an invariant approximation theorem is also derived.


2005 ◽  
Vol 47 (3-4) ◽  
pp. 242-265 ◽  
Author(s):  
Piroska Csörgő ◽  
Aleš Drápal

2003 ◽  
Vol 74 (3) ◽  
pp. 295-312 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev

AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.


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