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 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 SOME ALGEBRAIC AND TOPOLOGICAL PROPERTIES OF THE NONABELIAN TENSOR PRODUCT

2013 ◽  
Vol 50 (4) ◽  
pp. 1069-1077 ◽  
Author(s):  
Daniele Ettore Otera ◽  
Francesco G. Russo ◽  
Corrado Tanasi
Keyword(s):  
Tensor Product ◽  
Topological Properties ◽  
Nonabelian Tensor

The Homological Functors of Some Abelian Groups of Prime Power Order

2014 ◽  
Vol 71 (5) ◽  
Author(s):  
Rosita Zainal ◽  
Nor Muhainiah Mohd Ali ◽  
Nor Haniza Sarmin ◽  
Samad Rashid
Keyword(s):  
Tensor Product ◽  
Prime Power ◽  
Homotopy Theory ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Schur Multiplier ◽  
Integer Coefficients ◽  
Nonabelian Tensor ◽  
Tensor Square ◽  
Special Case

The homological functors of a group were first introduced in homotopy theory. Some of the homological functors including the nonabelian tensor square and the Schur multiplier of abelian groups of prime power order are determined in this paper. The nonabelian tensor square of a group G introduced by Brown and Loday in 1987 is a special case of the nonabelian tensor product. Meanwhile, the Schur multiplier of G is the second cohomology with integer coefficients is named after Issai Schur. The aims of this paper are to determine the nonabelian tensor square and the Schur multiplier of abelian groups of order p5, where p is an odd prime

scholarly journals Decomposition of the Nonabelian Tensor Product of Lie Algebras via the Diagonal Ideal

2017 ◽  
Vol 42 (4) ◽  
pp. 1295-1304
Author(s):  
Peyman Niroomand ◽  
Farangis Johari ◽  
Mohsen Parvizi ◽  
Francesco G. Russo
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Nonabelian Tensor

scholarly journals On the non-albelian tensor square of a nilpotent group of class two

1994 ◽  
Vol 36 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Michael R. Bacon
Keyword(s):  
Tensor Product ◽  
Nilpotent Group ◽  
Homotopy Theory ◽  
Nonabelian Tensor ◽  
Image Position ◽  
Tensor Square ◽  
Special Case

The nonabelian tensor square G⊗G of a group G is generated by the symbols g⊗h, g, h ∈ G, subject to the relations,for all g, g′, h, h′ ∈ G, where The tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J. L. Loday in [4] and [5], extending ideas of Whitehead in [6].

scholarly journals Two generalizations of the nonabelian tensor product

2012 ◽  
Vol 369 ◽  
pp. 96-113 ◽  
Author(s):  
M. Ladra ◽  
V.Z. Thomas
Keyword(s):  
Tensor Product ◽  
Nonabelian Tensor

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 SOME CONSIDERATIONS ON THE NONABELIAN TENSOR SQUARE OF CRYSTALLOGRAPHIC GROUPS

2011 ◽  
Vol 04 (02) ◽  
pp. 271-282 ◽  
Author(s):  
Ahmad Erfanian ◽  
Francesco G. Russo ◽  
Nor Haniza Sarmin
Keyword(s):  
Tensor Product ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Schur Multiplier ◽  
Crystallographic Groups ◽  
Nonabelian Tensor ◽  
Tensor Square

The nonabelian tensor square G ⊗ G of a polycyclic group G is a polycyclic group and its structure arouses interest in many contexts. The same assertion is still true for wider classes of solvable groups. This motivated us to work on two levels in the present paper: on a hand, we investigate the growth of the Hirsch length of G ⊗ G by looking at that of G, on another hand, we study the nonabelian tensor product of pro–p–groups of finite coclass, which are a remarkable class of solvable groups without center, and then we do considerations on their Hirsch length. Among other results, restrictions on the Schur multiplier will be discussed.

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