scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

Galois Extensions as Modules over the Group Ring

1970 ◽  
Vol 22 (2) ◽  
pp. 242-248 ◽  
Author(s):  
Gerald Garfinkel ◽  
Morris Orzech
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Galois Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Finite Abelian Group ◽  
Galois Extensions ◽  
Normal Bases ◽  
Direct Sum ◽  
Definition Of

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

ON THE UNITS OF THE INTEGRAL GROUP RING Z[G × Cp]

2008 ◽  
Vol 07 (03) ◽  
pp. 393-403 ◽  
Author(s):  
RICHARD M. LOW
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Direct Product ◽  
Group Ring ◽  
Open Problem ◽  
Structure Theorem ◽  
Integral Group Ring ◽  
Integral Group ◽  
Group Of Units ◽  
A Finite Group

Describing the group of units U(ZG) of the integral group ring ZG, for a finite group G, is a classical and open problem. In this paper, it is shown that U(Z[G × Cp]) = M ⋊ U(ZG), a semi-direct product where M is a certain subgroup of U(Z[ζ]G) and p prime. For p = 2, this structure theorem is applied to give precise descriptions of U(ZG) for a non-abelian group G of order 32, G = C10, and G = C8 × C2.

scholarly journals A connection between the number of subgroups and the order of a finite group

2020 ◽  
pp. 2250001
Author(s):  
Mihai-Silviu Lazorec
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Subgroup Lattice ◽  
Minimum Value ◽  
A Finite Group

For a finite group [Formula: see text], we associate the quantity [Formula: see text], where [Formula: see text] is the subgroup lattice of [Formula: see text]. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of [Formula: see text] on the class of [Formula: see text]-groups of order [Formula: see text], where [Formula: see text] is an integer. We show that the set containing the quantities [Formula: see text], where [Formula: see text] is a finite (abelian) group, is dense in [Formula: see text] Finally, we consider [Formula: see text] to be a function on [Formula: see text] and we indicate some of its properties, the main result being the classification of finite abelian [Formula: see text]-groups [Formula: see text] satisfying [Formula: see text]

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.

The Group of Units of the Integral Group Ring ZS3

1972 ◽  
Vol 15 (4) ◽  
pp. 529-534 ◽  
Author(s):  
I. Hughes ◽  
K. R. Pearson
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Torsion Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
List Type ◽  
Type Number ◽  
Quaternion Group ◽  
Group Of Units

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is(i) abelian and the order of each element divides 4, or(ii) abelian and the order of each element divides 6, or(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

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.

A Remark on the Units of Finite Order in The Group Ring of a Finite Group

1974 ◽  
Vol 17 (1) ◽  
pp. 129-130 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Finite Group ◽  
Finite Order ◽  
Direct Product ◽  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
Quaternion Group ◽  
Group Of Units ◽  
A Finite Group

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will proveLet G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

BREAKING UP FINITE AUTOMATA PRESENTABLE TORSION-FREE ABELIAN GROUPS

2011 ◽  
Vol 21 (08) ◽  
pp. 1463-1472 ◽  
Author(s):  
GÁBOR BRAUN ◽  
LUTZ STRÜNGMANN
Keyword(s):  
Abelian Group ◽  
Finite Automaton ◽  
Free Abelian Group ◽  
Finite Automata ◽  
Additive Group ◽  
Rational Numbers ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

In [Todor Tsankov, The additive group of the rationals does not have an automatic presentation, May 2009, arXiv:0905.1505v1], it was shown that the group of rational numbers is not FA-presentable, i.e. it does not admit a presentation by a finite automaton. More generally, any torsion-free abelian group that is divisible by infinitely many primes is not of this kind. In this article we extend the result from [13] and prove that any torsion-free FA-presentable abelian group G is an extension of a finite rank free group by a finite direct sum of Prüfer groups ℤ(p∞).

scholarly journals Groups containing locally maximal product-free sets of size 4

2021 ◽  
Vol 31 (2) ◽  
pp. 167-194
Author(s):  
C. S. Anabanti ◽  
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Open Problem ◽  
Finite Abelian Group ◽  
Locally Maximal ◽  
A Finite Group ◽  
Free Set ◽  
Free Sets

Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |S−−√|≤2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case.

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