Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3
TL;DR: In this paper, it was shown that if G is a nonabelian semidirect product of the form C p ⋊ C 3 k, with prime p > 7 and the cyclic 3-group acting like a group of order 3, then Z [ G ] does not have MJD.
About: This article is published in Journal of Algebra.The article was published on 2013-08-15 and is currently open access. It has received 4 citations till now. The article focuses on the topics: Group ring & Semidirect product.
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TL;DR: In this paper, it was shown that the integral group ring Z [ Q 8 × C p ] has the multiplicative Jordan decomposition property when m is congruent to 2 modulo 4.
1 citations
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TL;DR: In this article, the authors studied the nilpotent decomposition property in integral group rings and classified finite SSN groups such that the rational group algebra has only one Wedderburn component which is not a division ring.
Abstract: A finite group $G$ is said to have the nilpotent decomposition property (ND) if for every nilpotent element $\alpha$ of the integral group ring $\mathbb{Z}[G]$ one has that $\alpha e$ also belong to $\mathbb{Z}[G]$, for every primitive central idempotent $e$ of the rational group algebra $\mathbb{Q}[G]$. Results of Hales, Passi and Wilson, Liu and Passman show that this property is fundamental in the investigations of the multiplicative Jordan decomposition of integral group rings. If $G$ and all its subgroups have ND then Liu and Passman showed that $G$ has property SSN, that is, for subgroups $H$, $Y$ and $N$ of $G$, if $N\lhd H $ and $Y\subseteq H$ then $N\subseteq Y$ or $YN$ is normal in $H$; and such groups have been described. In this article, we study the nilpotent decomposition property in integral group rings and we classify finite SSN groups $G$ such that the rational group algebra $\mathbb{Q}[G]$ has only one Wedderburn component which is not a division ring.
TL;DR: In this article, the authors studied the nilpotent decomposition property in integral group rings and classified finite SSN groups G such that the rational group algebra Q [ G ] has only one Wedderburn component which is not a division ring.
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Book•
01 Jan 1938
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
Abstract: This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford, Cambridge, Aberdeen, and other universities. It is neither a systematic treatise on the theory of numbers nor a 'popular' book for non-mathematical readers. It contains short accounts of the elements of many different sides of the theory, not usually combined in a single volume; and, although it is written for mathematicians, the range of mathematical knowledge presupposed is not greater than that of an intelligent first-year student. In this edition, the main changes are in the notes at the end of each chapter. Sir Edward Wright seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present, both in the notes and in the text, a reasonably accurate account of the present state of knowledge.
5,972 citations
Book•
01 Jan 1960
TL;DR: Divisibility congruence quadratic reciprocity and Quadratic forms some functions of number theory some diophantine equations Farey fractions and irrational numbers simple continued fractions primes and multiplicative number theory algebraic numbers the partition function the density of sequences of integers.
Abstract: Divisibility congruence quadratic reciprocity and quadratic forms some functions of number theory some diophantine equations Farey fractions and irrational numbers simple continued fractions primes and multiplicative number theory algebraic numbers the partition function the density of sequences of integers.
1,508 citations
12 Dec 1991
TL;DR: In this article, Algebraic foundations and Dedekind domains have been used to define classes and units of low degree fields of cyclotomic fields and Diophantine equations.
Abstract: Notation Introduction 1 Algebraic foundations 2 Dedekind domains 3 Extensions 4 Classgroups and units 5 Fields of low degree 6 Cyclotomic fields 7 Diophantine equations 8 L-functions Appendices Exercises Glossary of theorems Index
671 citations
TL;DR: In this article, the authors studied the problem of unit formation in the ring of rational integers, where the identity of an element in G is always represented by a symbol e. The symbol e, with or without subscripts, will always denote an element.
Abstract: when addition and multiplication are defined in the obvious way, form a ring, the group-ring of G over K, which will be denoted by R (G, K). Henceforward, we suppose that K has the modulus 1, and we denote the identity in G by e0. Then R(G,K) has the modulus l.e0. Since no confusion can arise thereby, the element 1. e in R(G, K) will be written as e, and whenever it is convenient, the elements e0 in G and re0 in R(G, K) as 1 and r respectively. The symbol e, with or without subscripts, will always denote an element in G. If the elements Ex, E2 in R(G, K) satisfy JE1E2 = 1, Ex will be said to be a left unit, and E9 a right unit in R(G, K). If 77 is a left (or right) unit in K, then rje is a left (or right) unit in R(G, K). Such a unit will be described as trivial. The units in R(G, K) form a group if and only if every right unit is also a left unit. This is so, for instance, if both G and K are Abelian. It is also true if G is a finite group, and K is any ring of complex numbers, for then the regular representation| of G can be extended to give an isomorphism of R(G, K) in the ring of ordinary matrices. The first object of this paper is to study units in R(G,C), where C is the ring of rational integers. In § 2 we take G to be a finite Abelian group,
449 citations
129 citations