scholarly journals On the structure of a real crossed group algebra

1990 ◽  
Vol 41 (1) ◽  
pp. 113-115
Author(s):  
P.P. Pálfy
Keyword(s):  
Left Ideal ◽  
Group Algebra ◽  
Finite Index ◽  
Cyclic Subgroup ◽  
Dihedral Group ◽  
Real Field ◽  
Finitely Generated ◽  
The Real ◽  
Torsion Free

We prove that the crossed group algebra A of the infinite dihedral group over the real field defined by the generators a and b, relations b−1 ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ is a principal left ideal ring. This corrects a result of Buzási and provides the missing step towards the classification of finitely generated torsion-free RG-modules for groups G which contain an infinite cyclic subgroup of finite index.

scholarly journals On the structure of a real crossed group algebra

1988 ◽  
Vol 38 (1) ◽  
pp. 31-40 ◽  
Author(s):  
K. Buzási
Keyword(s):  
Group Algebra ◽  
Dihedral Group ◽  
Matrix Algebras ◽  
Generating Sets ◽  
Zero Divisors ◽  
Finite Set ◽  
Representations Of Finite Groups ◽  
New Ideas ◽  
Real Quaternions ◽  
Torsion Free

The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebra A of the infinite dihedral group ≤ a, b | b−1ab = a−1, b2 = 1 ≥ over the field R of real numbers: namely in the A defined by b−l ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ.The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groups G which have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the present A. The point of the main result is that new ideas will be needed for understanding A-modules.A number of subsidiary results are concerned with convenient generating sets for left ideals in A.

Faithful Representations of Finitely Generated Metabelian Groups

1975 ◽  
Vol 27 (6) ◽  
pp. 1355-1360 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Finite Index ◽  
Characteristic Zero ◽  
Metabelian Group ◽  
Finite Extension ◽  
Faithful Representation ◽  
Finitely Generated ◽  
Finite Degree ◽  
Metabelian Groups ◽  
Finite Exponent ◽  
Torsion Free

In [3] Remeslennikov proves that a finitely generated metabelian group G has a faithful representation of finite degree over some field F of characteristic zero (respectively, p > 0) if its derived group G’ is torsion-free (respectively, of exponent p). By the Lie-Kolchin-Mal'cev theorem any metabelian subgroup of GL(n, F) has a subgroup of finite index whose derived group is torsion-free if char F = 0 and is a p-group of finite exponent if char F = p > 0. Moreover every finite extension of a group with a faithful representation (of finite degree) has a faithful representation over the same field. Thus Remeslennikov's results have a gap which we propose here to fill.

A Finite Index Property of Certain Solvable Groups

1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith
Keyword(s):  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

scholarly journals ON THE CAPABILITY OF FINITELY GENERATED NON-TORSION GROUPS OF NILPOTENCY CLASS 2

2011 ◽  
Vol 53 (2) ◽  
pp. 411-417 ◽  
Author(s):  
LUISE-CHARLOTTE KAPPE ◽  
NOR MUHAINIAH MOHD ALI ◽  
NOR HANIZA SARMIN
Keyword(s):  
Torsion Group ◽  
Nilpotency Class ◽  
Necessary Condition ◽  
Finitely Generated ◽  
Free Rank ◽  
Torsion Groups ◽  
Necessary And Sufficient ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractA group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

scholarly journals Subgroups of free solvable groups

1971 ◽  
Vol 12 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Jacques Lewin ◽  
Tekla Lewin
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Finite Index ◽  
Solvable Groups ◽  
Free Subgroup ◽  
Infinite Index ◽  
Finitely Generated ◽  
Derived Length ◽  
Inner Automorphisms ◽  
Torsion Free

A consequence of Schreier's formula is that if G is a subgroup of the free group F of rank n < 1 and rank G ≦ n, then G = F or G is of infinite index in F. However, if S is a free sovlvable group of derived length I < 1 and H is a subgroup of S which is free solvable of the same length, then the rank of H does not exceed the rank of S. These observations led G. Baumslag to conjecture that if H is of finite index in S then H = S. In fact, we have sharper results in two directions. If H and S are free solvable of the same length, not only is H of infinite index in S, but δ1−1(S)/δ1−1(H) is torsion-free. In another direction we need not assume that S is free solvable, only that s is torsion-free and of derived length l (l > 1) and that H is not cyclic. Thus Stallings' theorem [11] that a finitely generated torsionfree group with a free subgroup of finite index is itself free has an even stronger counterpart in the variety of groups solvable of length at most l (l > 1): a torsionfree group in that variety with a non-cyclic free subgroup of finite index coincides with this subgroup. The proof relies on the following theorem: If S is a free solvable group, J is the group of automorphisms of S which induce the identity on S/S', and I is the group of inner automorphisms of S, then J/I is torsion-free. The proofs of these theorems form the bulk of the first four sections.

Infinite groups whose group algebras satisfy the converse of Schur’s lemma

2019 ◽  
Vol 18 (10) ◽  
pp. 1950186
Author(s):  
Mohammed El Badry ◽  
Mostafa Alaoui Abdallaoui ◽  
Abdelfattah Haily
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Abelian Subgroup ◽  
Sufficient Conditions ◽  
Group Algebras ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Soluble Groups ◽  
Schur’S Lemma ◽  
Schur's Lemma

In this work, we give some necessary and/or sufficient conditions for a group algebra of infinite group to satisfy the converse of Schur’s Lemma. Many classes of groups are investigated such as abelian groups, hypercentral groups, groups having abelian subgroup of finite index and finitely generated soluble groups.

Some Characterizations of a Normal Subgroup of a Group and Isotopic Classes of Transversals

2016 ◽  
Vol 23 (03) ◽  
pp. 409-422 ◽  
Author(s):  
Vipul Kakkar ◽  
R. P. Shukla
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Dihedral Group ◽  
Elementary Proof ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Right Loop

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this paper, we give some characterizations for the normality of H in G. As a consequence we get a very short and elementary proof of the main theorem of a paper of Lal and Shukla, which avoids the use of the classification of finite simple groups. Further, we study the isotopy between the transversals in some groups and determine the number of isotopy classes of transversals of a subgroup of order 2 in D2p, the dihedral group of order 2p, where p is an odd prime and the isotopism classes are formed with respect to induced right loop structures.

A note on a centrality property of finitely generated soluble groups

1974 ◽  
Vol 75 (1) ◽  
pp. 23-24 ◽  
Author(s):  
J. C. Lennox
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Finite Index ◽  
Finitely Generated ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
Finitely Generated Group ◽  
Torsion Free ◽  
Simpler Method

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.

Stochastic simulation of the spatio-temporal field of the average daily heat index in Southern Russia

2020 ◽  
Vol 82 ◽  
pp. 149-160
Author(s):  
N Kargapolova
Keyword(s):  
Heat Waves ◽  
Numerical Models ◽  
Heat Index ◽  
Real Field ◽  
Gaussian Field ◽  
The Real ◽  
Southern Russia ◽  
Weather Stations ◽  
Spatio Temporal ◽  
Temporal Field

Numerical models of the heat index time series and spatio-temporal fields can be used for a variety of purposes, from the study of the dynamics of heat waves to projections of the influence of future climate on humans. To conduct these studies one must have efficient numerical models that successfully reproduce key features of the real weather processes. In this study, 2 numerical stochastic models of the spatio-temporal non-Gaussian field of the average daily heat index (ADHI) are considered. The field is simulated on an irregular grid determined by the location of weather stations. The first model is based on the method of the inverse distribution function. The second model is constructed using the normalization method. Real data collected at weather stations located in southern Russia are used to both determine the input parameters and to verify the proposed models. It is shown that the first model reproduces the properties of the real field of the ADHI more precisely compared to the second one, but the numerical implementation of the first model is significantly more time consuming. In the future, it is intended to transform the models presented to a numerical model of the conditional spatio-temporal field of the ADHI defined on a dense spatio-temporal grid and to use the model constructed for the stochastic forecasting of the heat index.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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