scholarly journals On the structure of a real crossed group algebra

1988 ◽  
Vol 38 (1) ◽  
pp. 31-40 ◽  
Author(s):  
K. Buzási
Keyword(s):  
Group Algebra ◽  
Dihedral Group ◽  
Matrix Algebras ◽  
Generating Sets ◽  
Zero Divisors ◽  
Finite Set ◽  
Representations Of Finite Groups ◽  
New Ideas ◽  
Real Quaternions ◽  
Torsion Free

The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebra A of the infinite dihedral group ≤ a, b | b−1ab = a−1, b2 = 1 ≥ over the field R of real numbers: namely in the A defined by b−l ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ.The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groups G which have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the present A. The point of the main result is that new ideas will be needed for understanding A-modules.A number of subsidiary results are concerned with convenient generating sets for left ideals in A.

Zero divisors and idempotents in group rings

1977 ◽  
Vol 81 (3) ◽  
pp. 365-368 ◽  
Author(s):  
P. A. Linnell
Keyword(s):  
Group Algebra ◽  
Zero Divisor ◽  
Affirmative Answer ◽  
Supersoluble Group ◽  
Group Rings ◽  
Prime Characteristic ◽  
Locally Finite ◽  
Zero Divisors ◽  
Torsion Free

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

scholarly journals On the structure of a real crossed group algebra

1990 ◽  
Vol 41 (1) ◽  
pp. 113-115
Author(s):  
P.P. Pálfy
Keyword(s):  
Left Ideal ◽  
Group Algebra ◽  
Finite Index ◽  
Cyclic Subgroup ◽  
Dihedral Group ◽  
Real Field ◽  
Finitely Generated ◽  
The Real ◽  
Torsion Free

We prove that the crossed group algebra A of the infinite dihedral group over the real field defined by the generators a and b, relations b−1 ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ is a principal left ideal ring. This corrects a result of Buzási and provides the missing step towards the classification of finitely generated torsion-free RG-modules for groups G which contain an infinite cyclic subgroup of finite index.

Concentrating standing waves for Davey–Stewartson systems

2021 ◽  
pp. 1-40
Author(s):  
Yi He ◽  
Xiao Luo
Keyword(s):  
Water Waves ◽  
Elliptic Equations ◽  
Standing Waves ◽  
Regularity Theory ◽  
Local Minima ◽  
Shallow Water Waves ◽  
Penalization Method ◽  
Finite Set ◽  
New Ideas ◽  
Concentrating Solutions

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

On the order of unitary subgroup of the modular group algebra 𝔽2kD2N

2015 ◽  
Vol 14 (08) ◽  
pp. 1550129 ◽  
Author(s):  
Neha Makhijani ◽  
R. K. Sharma ◽  
J. B. Srivastava
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Dihedral Group ◽  
Group Of Units ◽  
Modular Group Algebra ◽  
Canonical Involution ◽  
Unitary Subgroup

Let 𝔽qD2N be the group algebra of D2N, the dihedral group of order 2N, over 𝔽q = GF (q). In this paper, we compute the order of the unitary subgroup of the group of units of 𝔽2kD2N with respect to the canonical involution ∗.

scholarly journals An embedding of the group algebra of a torsion-free one-relator group in a field

1978 ◽  
Vol 52 (1) ◽  
pp. 39-74 ◽  
Author(s):  
Jacques Lewin ◽  
Tekla Lewin
Keyword(s):  
Group Algebra ◽  
Torsion Free

scholarly journals IDEMPOTENT RANK IN THE ENDOMORPHISM MONOID OF A NONUNIFORM PARTITION

2015 ◽  
Vol 93 (1) ◽  
pp. 73-91 ◽  
Author(s):  
IGOR DOLINKA ◽  
JAMES EAST ◽  
JAMES D. MITCHELL
Keyword(s):  
Endomorphism Monoid ◽  
Generating Sets ◽  
Idempotent Rank ◽  
Finite Set ◽  
Uniform Case

We calculate the rank and idempotent rank of the semigroup ${\mathcal{E}}(X,{\mathcal{P}})$ generated by the idempotents of the semigroup ${\mathcal{T}}(X,{\mathcal{P}})$ which consists of all transformations of the finite set $X$ preserving a nonuniform partition ${\mathcal{P}}$. We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.

scholarly journals UNITS IN F2D2p

2013 ◽  
Vol 13 (02) ◽  
pp. 1350090 ◽  
Author(s):  
KULDEEP KAUR ◽  
MANJU KHAN
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Normal Subgroups ◽  
Canonical Involution ◽  
Unitary Subgroup ◽  
Bicyclic Units

Let p be an odd prime, D2p be the dihedral group of order 2p, and F2 be the finite field with two elements. If * denotes the canonical involution of the group algebra F2D2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group [Formula: see text], generated by the bicyclic units of the group algebra F2D2p. Further, we obtain the structure of the unit group [Formula: see text] and the unitary subgroup [Formula: see text], and we prove that both [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text].

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