scholarly journals Structure of group rings and the group of units of integral group rings: an invitation

2021 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units

Free Subgroups in the Unit Groups of Integral Group Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1342-1352 ◽  
Author(s):  
B. Hartley ◽  
P. F. Pickel
Keyword(s):  
Sufficient Conditions ◽  
Finite Subgroup ◽  
Free Subgroup ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
Necessary And Sufficient ◽  
Free Subgroups

Let G be a group, ZG the group ring of G over the ring Z of integers, and U(ZG) the group of units of ZG. One method of investigating U(ZG) is to choose some property of groups and try to determine the groups G such that U(ZG) enjoys that property. For example Sehgal and Zassenhaus [9] have given necessary and sufficient conditions for U(ZG) to be nilpotent (see also [7]), and the same authors have investigated when U(ZG) is an FC (finite-conjugate) group [10]. For a survey of related questions, see [3]. In this paper we consider when U(ZG) contains a free subgroup of rank 2. We conjecture that if this does not happen, then every finite subgroup of G is normal, from which various other conclusions then follow (see Lemma 4).

scholarly journals Integral Group Rings with Nilpotent Unit Groups

1976 ◽  
Vol 28 (5) ◽  
pp. 954-960 ◽  
Author(s):  
César Polcino Milies
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Unit Element ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
The Subject ◽  
A Finite Group

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.The study of the nilpotency of U(RG) has been the subject of several papers.

A SURVEY ON FREE SUBGROUPS IN THE GROUP OF UNITS OF GROUP RINGS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350004 ◽  
Author(s):  
JAIRO Z. GONÇALVES ◽  
ÁNGEL DEL RÍO
Keyword(s):  
Finite Groups ◽  
Free Groups ◽  
Group Algebras ◽  
Group Rings ◽  
Integral Group ◽  
Natural Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Survey Results ◽  
Free Subgroups

In this survey we revise the methods and results on the existence and construction of free groups of units in group rings, with special emphasis in integral group rings over finite groups and group algebras. We also survey results on constructions of free groups generated by elements which are either symmetric or unitary with respect to some involution and other results on which integral group rings have large subgroups which can be constructed with free subgroups and natural group operations.

Alternating units as free factors in the group of units of integral group rings

2011 ◽  
Vol 54 (3) ◽  
pp. 695-709 ◽  
Author(s):  
Jairo Z. Gonçalves ◽  
Paula M. Veloso
Keyword(s):  
Central Element ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Odd Order

AbstractLet G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in $\mathcal{U}(\mathbb{Z}G)$, the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that 〈um, vm〉 = 〈um〉 ∗ 〈vm〉 ≌ ℤ ∗ ℤ

Integral Group Rings of Some p-Groups

1982 ◽  
Vol 34 (1) ◽  
pp. 233-246 ◽  
Author(s):  
Jürgen Ritter ◽  
Sudarshan Sehgal
Keyword(s):  
Dihedral Group ◽  
Alternating Group ◽  
Group Rings ◽  
Integral Group ◽  
Product Decomposition ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Wedderburn Decomposition ◽  
Fibre Product

1. Introduction. The group of units, , of the integral group ring of a finite non-abelian group G is difficult to determine. For the symmetric group of order 6 and the dihedral group of order 8 this was done by Hughes-Pearson [3] and Polcino Milies [5] respectively. Allen and Hobby [1] have computed , where A4 is the alternating group on 4 letters. Recently, Passman-Smith [6] gave a nice characterization of where D2p is the dihedral group of order 2p and p is an odd prime. In an earlier paper [2] Galovich-Reiner-Ullom computed when G is a metacyclic group of order pq with p a prime and q a divisor of (p – 1). In this note, using the fibre product decomposition as in [2], we give a description of the units of the integral group rings of the two noncommutative groups of order p3, p an odd prime. In fact, for these groups we describe the components of ZG in the Wedderburn decomposition of QG.

Units in Integral Group Rings for Order pq

1995 ◽  
Vol 47 (1) ◽  
pp. 113-131
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Finite Index ◽  
Finite Abelian Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Cyclotomic Units

AbstractFor any finite abelian group A, let Ω(A) denote the group of units in the integral group ring which are mapped to cyclotomic units by every character of A. It always contains a subgroup Y(A), of finite index, for which a basis can be systematically exhibited. For A of order pq, where p and q are odd primes, we derive estimates for the index [Ω(A) : Y(A)]. In particular, we obtain conditions for its triviality.

Semigroup identities on units of integral group rings

1997 ◽  
Vol 39 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Jairo Z. Gonçalves
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Group Ring ◽  
Torsion Group ◽  
Group Rings ◽  
Integral Group ◽  
Semigroup Identity ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Semigroup Identities

AbstractLet U(RG) be the group of units of a group ring RG over a commutative ring R with 1. We say that a group is an SIT-group if it is an extension of a group which satisfies a semigroup identity by a torsion group. It is a consequence of the main result that if G is torsion and R = Z, then U(RG) is an SIT-group if and only if G is either abelian or a Hamiltonian 2-group. If R is a local ring of characteristic 0 only the first alternative can occur.

Coherent groups of units of integral group rings and direct products of free groups

2016 ◽  
Vol 162 (2) ◽  
pp. 191-209
Author(s):  
ÁNGEL DEL RÍO ◽  
PAVEL ZALESSKII
Keyword(s):  
Direct Product ◽  
Division Algebra ◽  
Finite Groups ◽  
Free Groups ◽  
Group Rings ◽  
Integral Group ◽  
Direct Products ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Finite Dimensional

AbstractWe classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$, the group of units of the integral group ring of G, does not contain a direct product of two non-abelian free groups. This list of groups contains all the groups for which $\mathcal{U}({\mathbb Z} G)$ is coherent. This reduces the problem to classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$ is coherent to decide about the coherency of a finite list of groups of the form SLn(R), with R an order in a finite dimensional rational division algebra.

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