scholarly journals On a simple method of determing the homology and the cohomology of finitely presented groups

1970 ◽  
Vol 34 (2) ◽  
pp. 103-114
Author(s):  
Subrata Majumdar ◽  
Nasima Akhter
Keyword(s):  
Linear Equations ◽  
Free Resolution ◽  
Nilpotent Groups ◽  
General Situation ◽  
Group Rings ◽  
Integral Group ◽  
Group Presentation ◽  
Simple Method ◽  
Finitely Presented Groups ◽  
Finitely Presented

In this paper the authors obtained a method of constructing free resolutions of Z for finitely presented groups directly from their presentations by extending Lyndon’s 3-term partial resolution to a full-length resolution. Authors resolutions and the method of their construction are such that free generators of the modules and the boundary homomorphisms are directly and explicitly obtained by solving of linear equations over the corresponding integral group rings, and hence these are immediately applicable for computing homology and cohomology of the groups for arbitrary coefficient modules. Authors have also described a general situation where their method is valid. The method has been used for a number of classes of group including Fuchsian groups, a few Euclidean crystallographic groups, NEC groups, the fundamental groups of a few interesting manifolds, groups of isometries of the hyperbolic plane and a few nilpotent groups of class 2. Key words: Group presentation; Free resolution; Homology; Cohomology DOI: 10.3329/jbas.v34i2.6854Journal of Bangladesh Academy of Sciences, Vol. 34, No. 2, 103-114, 2010

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

Finite Subgroups in Integral Group Rings

1996 ◽  
Vol 48 (6) ◽  
pp. 1170-1179 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Stanley O. Juriaans
Keyword(s):  
Solvable Groups ◽  
Nilpotent Groups ◽  
Fourth Power ◽  
Group Rings ◽  
Integral Group ◽  
Frobenius Groups ◽  
Integral Group Rings ◽  
Finite Subgroups ◽  
Generalized Quaternion

AbstractA p-subgroup version of the conjecture of Zassenhaus is proved for some finite solvable groups including solvable groups in which any Sylow p-subgroup is either abelian or generalized quaternion, solvable Frobenius groups, nilpotent-by-nilpotent groups and solvable groups whose orders are not divisible by the fourth power of any prime.

scholarly journals Global and local properties of finite groups with only finitely many central units in their integral group ring

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andreas Bächle ◽  
Mauricio Caicedo ◽  
Eric Jespers ◽  
Sugandha Maheshwary
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Galois Groups ◽  
Group Rings ◽  
Integral Group ◽  
Open Problems ◽  
Integral Group Rings ◽  
Local Properties ◽  
Global And Local ◽  
The Impact

Abstract The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group, we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same; in particular, the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.

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