scholarly journals On the First Zassenhaus Conjecture and Direct Products

2018 ◽  
Vol 72 (3) ◽  
pp. 602-624
Author(s):  
Andreas Bächle ◽  
Wolfgang Kimmerle ◽  
Mariano Serrano
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Frobenius Group ◽  
Finite Abelian Group ◽  
Hall Subgroup ◽  
Group Rings ◽  
Direct Products ◽  
Ring Of Algebraic Integers ◽  
Zassenhaus Conjecture ◽  
Sylow Tower

AbstractIn this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals Picard Groups and Refined Discrete Logarithms

2005 ◽  
Vol 8 ◽  
pp. 1-16 ◽  
Author(s):  
W. Bley ◽  
M. Endres
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Finite Abelian Group ◽  
Discrete Logarithm ◽  
Quadratic Number ◽  
Prime Degree ◽  
Quadratic Number Field ◽  
Picard Groups ◽  
Imaginary Quadratic Number ◽  
Ring Of Algebraic Integers

AbstractLet K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

A characteristic property of elementary 2-groups of rank six

2011 ◽  
Vol 48 (3) ◽  
pp. 354-370
Author(s):  
Sándor Szabó
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Characteristic Property ◽  
Identity Element ◽  
Finite Abelian Group

Consider a finite abelian group G which is a direct product of its subsets A and B both containing the identity element e. If the non-periodicity of A and B forces that neither A nor B can span the whole G, then G must be an elementary 2-group of rank six.

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 Warfield p-Invariants in Abelian Group Rings of Characteristic p

2013 ◽  
Vol 31 (2) ◽  
pp. 183
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Group Ring ◽  
General Strategy ◽  
Group Rings ◽  
Finitely Generated ◽  
Characteristic P ◽  
Commutative Group Ring ◽  
Normalized Units ◽  
Finite Direct Product

We calculate Warfield p-invariants Wα,p(V (RG)) of the group of normalized units V (RG) in a commutative group ring RG of prime char(RG) = p in each of the following cases: (1) G0/Gp is finite and R is an arbitrary direct product of indecomposable rings; (2) G0/Gp is bounded and R is a finite direct product of fields; (3) id(R) is finite (in particular, R is finitely generated). Moreover, we give a general strategy for the computation of the above Warfield p-invariants under some restrictions on R and G. We also point out an essential incorrectness in a recent paper due to Mollov and Nachev in Commun. Algebra (2011).

Units of commutative group rings over polynomial ring

2018 ◽  
Vol 13 (01) ◽  
pp. 2050021
Author(s):  
S. Kaur ◽  
M. Khan
Keyword(s):  
Abelian Group ◽  
Finite Field ◽  
Polynomial Ring ◽  
Group Algebra ◽  
Modular Group ◽  
Finite Abelian Group ◽  
Unit Group ◽  
Group Rings ◽  
Modular Group Algebra ◽  
Univariate Polynomial

In this paper, we obtain the structure of the normalized unit group [Formula: see text] of the modular group algebra [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is the univariate polynomial ring over a finite field [Formula: see text] of characteristic [Formula: see text]

Direct products of subsets in a finite abelian group

2012 ◽  
Vol 138 (3) ◽  
pp. 226-236
Author(s):  
Keresztély Corrádi ◽  
Sándor Szabó
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Direct Products

10. Vector spaces

2015 ◽  
pp. 126-137
Author(s):  
Peter M. Higgins
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Fields ◽  
Finite Abelian Group ◽  
Scalar Multiplication ◽  
Modular Arithmetic ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Cyclic Groups ◽  
Factorization Of Polynomials

‘Vector spaces’ discusses the algebra of vector spaces, which are abelian groups with an additional scalar multiplication by a field. Every finite abelian group is the direct product of cyclic groups. Any finite abelian group can be represented in one of two special ways based on numerical relationships between the subscripts of the cyclic groups involved. In one representation, all the subscripts are powers of primes; in the alternative, each subscript is a divisor of its successor. It concludes by bringing together the ideas of modular arithmetic, the construction of the complex numbers, factorization of polynomials, and vector spaces to explain the existence of finite fields.

scholarly journals On finite loops whose inner mapping groups are Abelian II

2005 ◽  
Vol 71 (3) ◽  
pp. 487-492
Author(s):  
Markku Niemenmaa
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Inner Mapping Group ◽  
Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

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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