The tensor product and the 2nd nilpotent product of groups

1960 ◽  
Vol 73 (2) ◽  
pp. 134-145 ◽  
Author(s):  
Trueman MacHenry
Keyword(s):  
Tensor Product ◽  
Nilpotent Product ◽  
Product Of Groups

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.

Non-abelian tensor and exterior products of multiplicative Lie rings

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Guram Donadze ◽  
Nick Inassaridze ◽  
Manuel Ladra
Keyword(s):  
Tensor Product ◽  
Lie Rings ◽  
Product Of Groups

AbstractWe define a non-abelian tensor product of multiplicative Lie rings. This is a new concept providing a common approach to the non-abelian tensor product of groups defined by Brown and Loday and to the non-abelian tensor product of Lie rings defined by Ellis. We also prove an analogue of Miller’s theorem for multiplicative Lie rings.

scholarly journals Finiteness conditions for the non-abelian tensor product of groups

2017 ◽  
Vol 187 (4) ◽  
pp. 603-615 ◽  
Author(s):  
R. Bastos ◽  
I. N. Nakaoka ◽  
N. R. Rocco
Keyword(s):  
Tensor Product ◽  
Finiteness Conditions ◽  
Product Of Groups

scholarly journals A non-abelian tensor product of Lie algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 101-120 ◽  
Author(s):  
Graham J. Ellis
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Algebraic Theory ◽  
Tensor Products ◽  
Algebraic Structures ◽  
Commutative Algebras ◽  
Group Case ◽  
Do So ◽  
Product Of Groups

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

scholarly journals The tensor product of distributive lattices

1976 ◽  
Vol 20 (2) ◽  
pp. 121-131 ◽  
Author(s):  
Grant A. Fraser
Keyword(s):  
Tensor Product ◽  
Distributive Lattices ◽  
Variety Of Semigroups ◽  
Algebraic Concept ◽  
Product Of Groups

The well-known algebraic concept of tensor product exists for any variety of algebras.The tensor product of groups and of rings have been studied extensively. For other varieties, such as the variety of semigroups, the tensor product has been investigated more recently (5). In this paper we investigate the tensor product of distributive lattices.

On the Non-abelian Tensor Product of Groups

2011 ◽  
Vol 18 (03) ◽  
pp. 429-436
Author(s):  
Mohammad Reza R. Moghaddam ◽  
Fateme Mirzaei
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Nilpotent Group ◽  
Upper Bound ◽  
Upper Bounds ◽  
Schur Multiplier ◽  
Product Of Groups

In this paper, we study some properties of the non-abelian tensor product of two groups G and H. More precisely, if G is abelian and H is a nilpotent group, then an upper bound for the exponent of G ⊗ H is obtained. Using our results, we obtain some upper bounds for the exponent of the Schur multiplier of the non-abelian tensor product of groups. Finally, an abelian group is constructed by taking non-abelian tensor product of groups.

Some structural and closure properties of an extension of the q-tensor product of groups, q≥0

2021 ◽  
pp. 1-14
Author(s):  
Ivonildes Ribeiro Martins Dias ◽  
Noraí Romeu Rocco ◽  
Eunice Cândida Pereira Rodrigues
Keyword(s):  
Tensor Product ◽  
Closure Properties ◽  
Product Of Groups

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.   

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