Units in Group Rings of the Infinite Dihedral Group

1991 ◽  
Vol 34 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Maciej Mirowicz
Keyword(s):  
Group Ring ◽  
Integral Domain ◽  
Dihedral Group ◽  
Group Rings ◽  
Finitely Generated ◽  
Group Of Units ◽  
Group D

AbstractThis paper studies the group of units U(RD∞) of the group ring of the infinite dihedral group D∞ over a commutative integral domain R. The structures of U(Z2D∞) and U(Z3D∞) are determined, and it is shown that U(ZD∞) is not finitely generated.

Functors on Finite Vector Spaces and Units in Abelian Group Rings

1986 ◽  
Vol 29 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Elementary Abelian Group ◽  
Vector Spaces ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
Group Of Units ◽  
Finite Vector ◽  
Finite Vector Spaces

AbstractIf A is an elementary abelian group, let (A) denote the group of units, modulo torsion, of the group ring Z[A]. We study (A) by means of the compositewhere C and B run over all cyclic subgroups and factor-groups, respectively. This map, γ, is known to be injective; its index, too, is known. In this paper, we determine the rank of γ tensored (over Z);with various fields. Our main result depends only on the functoriality of

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.

scholarly journals Units of group rings of groups of order 16

1993 ◽  
Vol 35 (3) ◽  
pp. 367-379 ◽  
Author(s):  
E. Jespers ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Abelian Groups ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Group Of Units ◽  
Abelian Case ◽  
A Finite Group

LetGbe a finite group,(ZG) the group of units of the integral group ring ZGand1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively(ZG) for particular groupsG.This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described(ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of(ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

Free Subgroups of Units in Group Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 309-312 ◽  
Author(s):  
Jairo Zacarias Gonçalves
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Free Subgroup ◽  
Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Free Subgroups

AbstractIn this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.Some extensions of this results to infinite nilpotent and FC groups are also considered.

scholarly journals Public Key Protocols over Twisted Dihedral Group Rings

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1019
Author(s):  
María Dolores Gómez Olvera ◽  
Juan Antonio López Ramos ◽  
Blas Torrecillas Jover
Keyword(s):  
Information Security ◽  
Group Ring ◽  
Key Management ◽  
Dihedral Group ◽  
Complexity Analysis ◽  
Public Key ◽  
Group Rings ◽  
Decomposition Problem ◽  
Hard Problems ◽  
Twisted Group Ring

Key management is a central problem in information security. The development of quantum computation could make the protocols we currently use unsecure. Because of that, new structures and hard problems are being proposed. In this work, we give a proposal for a key exchange in the context of NIST recommendations. Our protocol has a twisted group ring as setting, jointly with the so-called decomposition problem, and we provide a security and complexity analysis of the protocol. A computationally equivalent cryptosystem is also proposed.

Group Rings with Units of Bounded Exponent over the Center

1982 ◽  
Vol 34 (6) ◽  
pp. 1349-1364 ◽  
Author(s):  
Sônia P. Coelho
Keyword(s):  
Solvable Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Group Rings ◽  
Group Of Units ◽  
Complete Answer ◽  
Group A

Let KG be the group ring of a group G over a field K, and U its group of units. Given a group H, we shall denote by ξ(H) the center of H and by T(H) the set of all its torsion elements.The following question appears in [5, p. 231]: When is Un ⊂ ξ (U), for some n? It was considered by G. Cliff and S. K. Sehgal in [1], where G is assumed to be a solvable group. A complete answer at characteristic zero is given there. Also they obtain partial results at characteristic p ≠ 0, with certain restrictions on the exponent n.

On Duo Group Rings

2011 ◽  
Vol 18 (01) ◽  
pp. 163-170
Author(s):  
Weidong Gao ◽  
Yuanlin Li
Keyword(s):  
Group Ring ◽  
Characteristic Zero ◽  
Integral Domain ◽  
Torsion Group ◽  
Rational Numbers ◽  
Group Rings ◽  
Quaternion Group ◽  
Quotient Field ◽  
Ring Of Algebraic Integers ◽  
Field Integral

It is shown that if the group ring RQ8 of the quaternion group Q8 of order 8 over an integral domain R is duo, then R is a field for the following cases: (1) char R ≠ 0, and (2) char R = 0 and S ⊆ R ⊆ KS, where S is a ring of algebraic integers and KS is its quotient field. Hence, we confirm that the field ℚ of rational numbers is the smallest integral domain R of characteristic zero such that RQ8 is duo. A non-field integral domain R of characteristic zero for which RQ8 is duo is also identified. Moreover, we give a description of when the group ring RG of a torsion group G is duo.

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.

scholarly journals Normal and subnormal subgroups in the group of units of group rings

1985 ◽  
Vol 31 (3) ◽  
pp. 355-363 ◽  
Author(s):  
Jairo Zacarias Goncalves
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Finite Subgroup ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Infinite Field ◽  
Group Of Units ◽  
Necessary And Sufficient ◽  
Subnormal Subgroups

Let KG be the group ring of the group G over the infinite field K, and let U(KG) be its group of units. If G is torsion, we obtain necessary and sufficient conditions for a finite subgroup H of G to be either normal or subnormal in U(KG). Actually, if H is subnormal in U(KG), we can handle not only the case H finite, but the precise assumptions depend on the characteristic of K.

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