A tensor product approach to compute 2-nilpotent multiplier of p-groups

2021 ◽  
pp. 2250090
Author(s):  
F. Fasihi ◽  
S. Hadi Jafari
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Product Formula ◽  
Exterior Product ◽  
Special Formula ◽  
Representations Of Groups ◽  
Nonabelian Tensor ◽  
Product Approach ◽  
Quotient Groups

Let [Formula: see text] be a group given by a free presentation [Formula: see text]. The 2-nilpotent multiplier of [Formula: see text] is the abelian group [Formula: see text] which is invariant of [Formula: see text] [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc. 58 (1945) 295–419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z. 226 (1997) 405–428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of [Formula: see text], when [Formula: see text] is a finite (generalized) extra special [Formula: see text]-group. Moreover, the descriptions of the triple tensor product [Formula: see text], and the triple exterior product [Formula: see text] are given.

scholarly journals On the Tensor Convolution and the Quantum Separability Problem

2010 ◽  
Vol 17 (04) ◽  
pp. 331-346
Author(s):  
Gabriel Pietrzkowski
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Hermitian Operator ◽  
Product Formula ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Equivalent Problem ◽  
Finite Dimensional ◽  
Separability Problem ◽  
Convolution Formula

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product [Formula: see text] is separable or entangled. We show that the tensor convolution [Formula: see text] defined for mappings [Formula: see text] on an almost arbitrary locally compact abelian group G , gives rise to formulation of an equivalent problem to the separability one.

A FEW HOMOLOGICAL CHARACTERIZATIONS OF COMPACT SEMISIMPLE RINGS

2005 ◽  
Vol 04 (05) ◽  
pp. 539-549
Author(s):  
ALINA ALB ◽  
MIHAIL URSUL
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Compact Abelian Group ◽  
Product Formula ◽  
Flat Module ◽  
Compact Ring ◽  
Module Theory ◽  
Flat Modules ◽  
Semisimple Module

Fix any compact ring R with identity. We associate to R the following categories of topological R-modules: (i) R𝔇 (𝔇R) the category of all discrete topological left (right) R-modules; (ii) Rℭ (ℭR) the category of all compact left (right) R-modules. We have introduced the following notions (analogous with classical notions of module theory): (i) the tensor product [Formula: see text] of A ∈ ℭR and B ∈Rℭ ([Formula: see text] has a structure of a compact Abelian group); (ii) a topologically semisimple module; (iii) a compact topologically flat module. We give a characterization of compact semisimple rings by using of flat modules.

scholarly journals Completely bounded bimodule maps and spectral synthesis

2017 ◽  
Vol 28 (10) ◽  
pp. 1750067 ◽  
Author(s):  
M. Alaghmandan ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Closed Subset ◽  
Spectral Synthesis ◽  
Locally Compact Group ◽  
Product Formula ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

scholarly journals Sequential completeness of quotient groups

2000 ◽  
Vol 61 (1) ◽  
pp. 129-150 ◽  
Author(s):  
Dikran Dikranjan ◽  
Michael Tkačenko
Keyword(s):  
Abelian Group ◽  
Direct Products ◽  
Topological Groups ◽  
Complete Group ◽  
Sequential Completeness ◽  
Algebraic Properties ◽  
Countable Compactness ◽  
Quotient Groups

We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

Semi-tensor product approach to networked evolutionary games

2014 ◽  
Vol 12 (2) ◽  
pp. 198-214 ◽  
Author(s):  
Daizhan Cheng ◽  
Hongsheng Qi ◽  
Fehuang He ◽  
Tingting Xu ◽  
Hairong Dong
Keyword(s):  
Tensor Product ◽  
Evolutionary Games ◽  
Product Approach

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.

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