scholarly journals The Homology and Cohomology Groups of H3

1970 ◽  
Vol 29 ◽  
pp. 139-146
Author(s):  
Subrata Majumdar ◽  
Quazi Selina Sultana
Keyword(s):  
Heisenberg Group ◽  
Key Words ◽  
Group Ring ◽  
Three Dimensional ◽  
Free Resolution ◽  
Integral Group Ring ◽  
Integral Group ◽  
Integral Homology ◽  
Cohomology Groups ◽  
Ams Classification

A free resolution of Z for the integral group ring of the three-dimensional Heisenberg group H3 has been constructed by extending Lyndon’s partial resolution. The integral homology and cohomology have been calculated from there.  AMS Classification: 18G, 20J. Key words: Fox derivatives; free resolution. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 139-146  DOI: http://dx.doi.org/10.3329/ganit.v29i0.8523 

scholarly journals Computation of Homology and cohomology groups of a metacyclic group and a factor group of the Heisenberg Group

2012 ◽  
Vol 31 ◽  
pp. 9-22
Author(s):  
Subrata Majumdar ◽  
Nasima Akter
Keyword(s):  
Heisenberg Group ◽  
Group Ring ◽  
Linear Equations ◽  
Factor Group ◽  
Free Resolutions ◽  
Integral Group ◽  
Metacyclic Group ◽  
Cohomology Groups ◽  
System Of Linear Equations ◽  
Free Generators

Free resolutions for a metacyclic group M and a factor group G of the Heisenberg group from their presentations constructed by solving system of linear equations over the integral group ring and determined the homology and cohomology groups. The method is straightforward, and the free resolutions are explicitly expressed in terms of the free generators. The resolutions yield the homology and the cohomology groups immediately.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10304GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 9-22

On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

1990 ◽  
Vol 42 (3) ◽  
pp. 383-394 ◽  
Author(s):  
Frank Röhl
Keyword(s):  
Group Ring ◽  
Modular Group ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Affirmative Answer ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

Trivial Units in Group Rings

2000 ◽  
Vol 43 (1) ◽  
pp. 60-62 ◽  
Author(s):  
Daniel R. Farkas ◽  
Peter A. Linnell
Keyword(s):  
Recent Result ◽  
Group Ring ◽  
Finite Index ◽  
Integral Group Ring ◽  
Alternative Proof ◽  
Crystallographic Group ◽  
Group Rings ◽  
Integral Group ◽  
Arbitrary Group ◽  
Normalized Units

AbstractLet G be an arbitrary group and let U be a subgroup of the normalized units in ℤG. We show that if U contains G as a subgroup of finite index, then U = G. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

scholarly journals INTEGRAL GROUP RING OF THE MATHIEU SIMPLE GROUP M24

2012 ◽  
Vol 11 (01) ◽  
pp. 1250016 ◽  
Author(s):  
VICTOR BOVDI ◽  
ALEXANDER KONOVALOV
Keyword(s):  
Simple Group ◽  
Group Ring ◽  
Unit Group ◽  
Positive Answer ◽  
Integral Group Ring ◽  
Integral Group ◽  
Sporadic Group ◽  
Prime Graphs ◽  
Zassenhaus Conjecture

We study the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M24. As a consequence, for this group we give a positive answer to the question by Kimmerle about prime graphs.

INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS

2011 ◽  
Vol 10 (04) ◽  
pp. 711-725 ◽  
Author(s):  
J. Z. GONÇALVES ◽  
D. S. PASSMAN
Keyword(s):  
Group Ring ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Nonabelian Group ◽  
Ring Of Integers ◽  
Integral Group Rings ◽  
Noncyclic Subgroup

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk, m(x), uk, m(x*)) or (uk, m(x), uk, m(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

The Hypercentre and the n-Centre of the Unit Group of an Integral Group Ring

1998 ◽  
Vol 50 (2) ◽  
pp. 401-411 ◽  
Author(s):  
Yuanlin Li
Keyword(s):  
Group Ring ◽  
Periodic Group ◽  
Unit Group ◽  
Complete Characterization ◽  
Integral Group Ring ◽  
Integral Group

AbstractIn this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most 2. We then give a complete characterization of the n-centre of that unit group. The n-centre of the unit group is either the centre or the second centre (for n ≥ 2).

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