Torsion units in integral group rings of some metabelian groups. II
type: Item
instance of: scholarly article
title: Torsion units in integral group rings of some metabelian groups. II
zbMATH Open document ID: 0611.16007
DOI: 10.1016/0022-314X(87)90037-0
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: This continuation of the work on the Zassenhaus conjecture [see Part I by the two last named authors, J. Algebra 103, 490-499 (1986; Zbl 0603.16010)] uses an entirely new approach which seems to be very interesting and, in the opinion of the authors, might lead to a proof of the Zassenhaus conjecture for groups G of the form \(A\rtimes X\) with A and X abelian and of relatively prime order. Instead of using explicit computations of the representations of G, the well known embedding \(ZG\to (ZA)_ m\) (m\(\times m\) matrices over ZA) is studied. For better application of this context, it is shown that the Zassenhaus conjecture is equivalent to the following statement: ''For every \(u=\sum u_ gg\), a torsion, normalized unit of ZG, there exists an element \(g_ 0\) of G, unique up to conjugation, such that the sum of the coefficients \(u_ g\) of the element g conjugate to \(g_ 0\) is different from zero''. With this new technique, the Zassenhaus conjecture is proved when the order of X, m, is less than the primes dividing the order of A in the following special cases: 1) m is prime; 2) A is cyclic.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3987492
zbMATH Keywords: torsion unit
zbMATH Keywords: integral group ring
zbMATH Keywords: group of units
zbMATH Keywords: Zassenhaus conjecture
zbMATH Keywords: embedding
zbMATH Keywords: normalized unit
author: Z. S. Marciniak
author: Jürgen Ritter
author: Sudarshan K. Sehgal
author: A. Weiss
reviewed by: Héctor A. Merklen
MaRDI profile type: Publication
cites work: Zero Divisors and Idempotents in Group Rings
cites work: Units of integral group rings of metabelian groups
cites work: On the centralizers of involutions in finite groups. II
cites work: Q5612629
cites work: On a Conjecture of Zassenhaus on Torsion Units in Integral Group Rings. II
cites work: Torsion units in integral group rings of metacyclic groups
cites work: On a conjecture of Zassenhaus on torsion units in integral group rings
cites work: Q4198893
cites work: Torsion units in integral group rings of some metabelian groups
full work available at URL: 0022-314x(87)90037-0
OpenAlex ID: W1974549909
Recommended article: Torsion units in integral group rings
Recommended article: Q3728170
Recommended article: Torsion units in matrix group rings
Recommended article: Zassenhaus conjecture for cyclic-by-Abelian groups.
Recommended article: Torsion Units in Integral Group Rings
Recommended article: TORSION UNITS IN INTEGRAL GROUP RINGS OF CERTAIN METABELIAN GROUPS
Recommended article: On the Torsion Units of Some Integral Group Rings
Recommended article: Integral group ring of the Mathieu simple group \(M_{12}\).
Recommended article: Zassenhaus conjecture on torsion units holds for \(\text{SL}(2, {p})\) and \(\text{SL}(2, {p}^2)\)
Recommended article: Integral group rings of frobenius groups and the conjectures of H.J. Zassenhaus
description: scientific article; zbMATH DE number 3987492