scholarly journals NUMBER OF COMPATIBLE PAIR OF ACTIONS FOR FINITE CYCLIC GROUPS OF P-POWER ORDER

2018 ◽  
Vol 80 (5) ◽  
Author(s):  
Mohammed Khalid Shahoodh ◽  
Mohd Sham Mohamad ◽  
Yuhani Yusof ◽  
Sahimel Azwal Sulaiman
Keyword(s):  
Tensor Product ◽  
Compatible Pair ◽  
Cyclic Groups ◽  
Exact Number ◽  
Nonabelian Tensor ◽  
Necessary And Sufficient ◽  
Product Of Groups

The compatible actions played an important role before determining the nonabelian tensor product of groups. Different compatible pair of actions gives a different nonabelian tensor product even for the same group. The aim of this paper is to determine the exact number of the compatible pair of actions for the finite cyclic groups of p-power order where p is an odd prime. By using the necessary and sufficient number theoretical conditions for a pair of the actions to be compatible with the actions that have p-power order, the exact number of the compatible pair of actions for the finite cyclic groups of p-power order has been determined and given as a main result in this paper.   

scholarly journals THE COMPUTATION OF THE NONABELIAN TENSOR PRODUCT OF CYCLIC GROUPS OF ORDER

2012 ◽  
Vol 57 (1) ◽  
Author(s):  
MOHD SHAM MOHAMAD ◽  
NOR HANIZA SARMIN ◽  
NOR MUHAINIAH MOHD ALI ◽  
LUISE–CHARLOTTE KAPPE
Keyword(s):  
Tensor Product ◽  
Cyclic Groups ◽  
Nonabelian Tensor

Katalah G dan H dua kumpulan yang bertindak ke atas satu sama lain dan masing-masing bertindak ke atasnya sendiri secara konjugasi, maka tindakan tersebut adalah serasi jika (gh)g' = g(h(g–1 g')) and for and . Keserasian tindakan adalah penting dalam penentuan hasil darab tensor tak abelan. Hasil darab tensor tak abelan, G⊗H telah diperkenalkan oleh Brown dan Loday pada tahun 1984. Hasil darab tensor tak abelan adalah kumpulan yang dijana oleh g⊗h dengan dua hubungan = dan g⊗hh' = (g⊗h)(hg⊗hh') bagi g,g'∈G dan h,h'∈Hdengan G dan H bertindak antara satu sama lain dalam tindakan yang serasi dan bertindak ke atas mereka sendiri dengan konjugasi. Pada tahun 1987, Brown dan rakan-rakan memberikan masalah terbuka dalam menentukan sama ada hasil darab tensor tak abelan dari dua kumpulan kitaran adalah juga kumpulan kitaran. Visscher pada tahun 1998 telah membuktikan bahawa hasil darab tensor tak abelan tidak semestinya kumpulan kitaran, tetapi kajian beliau hanya difokuskan kepada kumpulan kitaran berperingkat kuasa bagi dua dengan tindakannya berperingkat dua. Dalam makalah ini, keserasian dan hasil darab tensor tak abelan berperingkat p2 dengan tindakan berperingkat p ditentukan.(hg)h' = h(g(h–1 h'))g,g'∈Gh,h'∈Hgg'⊗h(gg'⊗gh)(g⊗h)

scholarly journals Product of the GeneralizedL-Subgroups

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Dilek Bayrak ◽  
Sultan Yamak
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Product Of Groups

We introduce the notion of(λ,μ)-product ofL-subsets. We give a necessary and sufficient condition for(λ,μ)-L-subgroup of a product of groups to be(λ,μ)-product of(λ,μ)-L-subgroups.

scholarly journals (r,p)-absolutely summing operators on the spaceC (T,X)and applications

2001 ◽  
Vol 6 (5) ◽  
pp. 309-315 ◽  
Author(s):  
Dumitru Popa
Keyword(s):  
Integral Operator ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Summing Operator ◽  
Absolutely Summing Operators ◽  
Injective Tensor Product ◽  
Absolutely Summing Operator ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for an operator on the spaceC (T,X)to be(r,p)-absolutely summing. Also we prove that the injective tensor product of an integral operator and an(r,p)-absolutely summing operator is an(r,p)-absolutely summing operator.

Finding Cut-Edges and the Minimum Spanning Tree via Semi-Tensor Product Approach

2018 ◽  
Vol 6 (5) ◽  
pp. 459-472
Author(s):  
Xujiao Fan ◽  
Yong Xu ◽  
Xue Su ◽  
Jinhuan Wang
Keyword(s):  
Tensor Product ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Product Approach ◽  
Matrix Expression ◽  
Necessary And Sufficient ◽  
Logical Functions

Abstract Using the semi-tensor product of matrices, this paper investigates cycles of graphs with application to cut-edges and the minimum spanning tree, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for cycles of graph, based on which a new necessary and sufficient condition is established to find all cycles for any graph. Secondly, using the necessary and sufficient condition of cycles, two algorithms are established to find all cut-edges and the minimum spanning tree, respectively. Finally, the study of an illustrative example shows that the results/algorithms presented in this paper are effective.

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.

A Family of Non-weight Modules over the Virasoro Algebra

2020 ◽  
Vol 27 (04) ◽  
pp. 807-820
Author(s):  
Guobo Chen
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Necessary And Sufficient Conditions ◽  
Weight Module ◽  
Highest Weight Module ◽  
Isomorphism Classes ◽  
Highest Weight ◽  
Necessary And Sufficient ◽  
Weight Modules

In this paper, we consider the tensor product modules of a class of non-weight modules and highest weight modules over the Virasoro algebra. We determine the necessary and sufficient conditions for such modules to be simple and the isomorphism classes among all these modules. Finally, we prove that these simple non-weight modules are new if the highest weight module over the Virasoro algebra is non-trivial.

scholarly journals Certain Groups of Orthonormal Step Functions

1957 ◽  
Vol 9 ◽  
pp. 413-425 ◽  
Author(s):  
J. J. Price
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Compact Abelian Group ◽  
Walsh Functions ◽  
Cyclic Groups ◽  
Step Functions ◽  
Product Of Groups

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

scholarly journals Compatible pair of actions for two same cyclic groups of 2-power order

2017 ◽  
Vol 890 ◽  
pp. 012120
Author(s):  
Mohd Sham Mohamad ◽  
Sahimel Azwal Sulaiman ◽  
Yuhani Yusof ◽  
Mohammed Khalid Shahoodh
Keyword(s):  
Compatible Pair ◽  
Cyclic Groups

A class of simple non-weight modules over the virasoro algebra

2020 ◽  
Vol 63 (4) ◽  
pp. 956-970 ◽  
Author(s):  
Haibo Chen ◽  
JianZhi Han
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Alpha Omega ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Lie Algebra

AbstractThe Virasoro algebra $\mathcal {L}$ is an infinite-dimensional Lie algebra with basis {Lm, C| m ∈ ℤ} and relations [Lm, Ln] = (n − m)Lm+n + δm+n,0((m3 − m)/12)C, [Lm, C] = 0 for m, n ∈ ℤ. Let $\mathfrak a$ be the subalgebra of $\mathcal {L}$ spanned by Li for i ≥ −1. For any triple (μ, λ, α) of complex numbers with μ ≠ 0, λ ≠ 0 and any non-trivial $\mathfrak a$-module V satisfying the condition: for any v ∈ V there exists a non-negative integer m such that Liv = 0 for all i ≥ m, non-weight $\mathcal {L}$-modules on the linear tensor product of V and ℂ[∂], denoted by $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))\ (\Omega (\lambda ,\alpha )=\mathbb {C}[\partial ]$ as vector spaces), are constructed in this paper. We prove that $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ is simple if and only if μ ≠ 1, λ ≠ 0, α ≠ 0. We also give necessary and sufficient conditions for two such simple $\mathcal {L}$-modules being isomorphic. Finally, these simple $\mathcal {L}$-modules $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ are proved to be new for V not being the highest weight $\mathfrak a$-module whose highest weight is non-zero.

scholarly journals The non-abelian tensor product of groups and related constructions

1989 ◽  
Vol 31 (1) ◽  
pp. 17-29 ◽  
Author(s):  
N. D. Gilbert ◽  
P. J. Higgins
Keyword(s):  
Tensor Product ◽  
Special Cases ◽  
Algebraic Description ◽  
Close Relationship ◽  
Closed Structure ◽  
Product Of Groups

The tensor product of two arbitrary groups acting on each other was introduced by R. Brown and J.-L. Loday in [5, 6]. It arose from consideration of the pushout of crossed squares in connection with applications of a van Kampen theorem for crossed squares. Special cases of the product had previously been studied by A. S.-T. Lue [10] and R. K. Dennis [7]. The tensor product of crossed complexes was introduced by R. Brown and the second author [3] in connection with the fundamental crossed complex π(X) of a filtered space X, which also satisfies a van Kampen theorem. This tensor product provides an algebraic description of the crossed complex π(X ⊗ Y) and gives a symmetric monoidal closed structure to the category of crossed complexes (over groupoids). Both constructions involve non-abelian bilinearity conditions which are versions of standard identities between group commutators. Since any group can be viewed as a crossed complex of rank 1, a close relationship might be expected between the two products. One purpose of this paper is to display the direct connections that exist between them and to clarify their differences.

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