Non-abelian Tensor Product Of Groups Research Articles

Non-Compatible Action Graph and Its Adjacency Matrix for The Non-abelian Tensor Product for Groups of Prime Power Order

This article focused on the notion of the non-abelian tensor product of groups of prime power order. Particularly, it presented new graph named as Non-compatible action graph and discussed some of its properties. Moreover, this graph concentrated on the case of non-compatible actions of the tensor product of two finite -groups. Furthermore, its adjacency matrix has been determined and discussed in detail. Moreover, the adjacency matrix has been denoted by A(G) and its inputs are 1 whenever there is adjacency and 0 otherwise.

Finiteness conditions for the box-tensor product of groups and related constructions

We study finiteness conditions and closure properties for the box-tensor product G⊠H and related constructions. The content of this paper extends some results concerning the non-abelian tensor product of groups G⊗H. In particular, we deduce a quantitative version of the finiteness criterion for the non-abelian tensor product. Moreover, we obtain finiteness conditions for some functors that arise out of the non-abelian tensor square of groups, such as the second homology group H2(G), the non-abelian exterior square G∧G and the second stable homotopy group of an Eilenberg-MacLane space π2S(K(G,1)).

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

In this work we study some structural properties of the group q a non-negative integer, which is an extension of the q-tensor product where G and H are normal subgroups of some group L. We establish by simple arguments some closure properties of when G and H belong to certain Schur classes. This extends similar results concerning the case q = 0 found in the literature. Restricting our considerations to the case G = H, we compute the q-tensor square for q odd, where Dn denotes the dihedral group of order 2n. Upper bounds for the exponent of are also established for nilpotent groups G of class which extend to all similar bounds found by Moravec, P. (2008). The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra. 212(7):1840–1848.

Compatible actions of Lie algebras

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object L from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in XModL(LieR).

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

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.

The Number of Compatible Pair of Actions for Cyclic Groups of 2-Power Order

The non-abelian tensor product of groups has its origins in the algebraic K-theory and homotopy theory. The nonabelian tensor product for a pair of groups is defined when the actions act compatibly on each other. This research is to determine the maximum number of a compatible pair of actions that can be identified between two cyclic groups of 2-power order for nonabelian tensor product. The compatible pair of actions between two cyclic groups of 2-power order can be found by using the necessary and sufficient conditions of two cyclic groups of 2-power order acting compatibly on each other. Hence, the number of the compatible pair of actions between two cyclic groups of the 2-power order is determined.

Finiteness conditions for the non-abelian tensor product of groups

Let $G$, $H$ be groups. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. We prove that if $G$ and $H$ are groups that act compatibly on each other and such that the set of all tensors $T_{\otimes}(G,H)=\{g\otimes h \, : \, g \in G, \, h\in H\}$ is finite, then the non-abelian tensor product $G \otimes H$ is finite. In the opposite direction we examine certain finiteness conditions of $G$ in terms of similar conditions for the tensor square $G \otimes G$.

Non-abelian tensor and exterior products of multiplicative Lie rings

Abstract We 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.

Two generalizations of the nonabelian tensor product

The purpose of this paper is two-fold. First we introduce the box-tensor product of two groups as a generalization of the nonabelian tensor product of groups. We extend various results for nonabelian tensor products to the box-tensor product such as the finiteness of the product when each factor is finite. This would give yet another proof of Ellisʼs theorem on the finiteness of the nonabelian tensor product of groups when each factor is finite. Secondly, using the methods developed in proving the finiteness of the box-tensor product, we prove the finiteness of Inassaridzeʼs tensor product under some additional hypothesis which generalizes his results on the finiteness of his product. In addition, we prove an Ellis like finiteness theorem under weaker assumptions, which is a generalization of his theorem on the finiteness of nonabelian tensor product. As a consequence, we prove the finiteness of low dimensional nonabelian homology groups.

A survey of non-abelian tensor products of groups and related constructions

We report on a group construction in connection with non-abelian tensor products of groups and recent development in non-abelian tensor products and q-tensor products.

On the Non-abelian Tensor 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.

The exponents of nonabelian tensor products of groups

We prove that the exponent of the nonabelian tensor product of two locally finite groups can be bounded in terms of exponents of given groups. Several estimates for the exponents of nonabelian tensor squares are obtained. In particular, if the group G is nilpotent of class ≤ 3 and of finite exponent, then the exponent of its nonabelian tensor square divides the exponent of G .

Universal central extensions of precrossed modules and Milnor's relative K2

We characterize the universal central extension of a perfect precrossed module giving two descriptions, one in terms of non-abelian tensor products of groups and other in terms of projective presentations. As application to relative algebraic K-theory, we obtain that Milnor's absolute and relative K 2 groups are the kernel of the universal central extension of the precrossed module determined by the groups of the elementary matrices of a ring and relative to an ideal, respectively.

On the relation between upper central quotients and lower central series of a group

Let H be a group with a normal subgroup N contained in the upper central subgroup ZcH. In this article we study the influence of the quotient group G = H/N on the lower central subgroup yc+lH. In particular, for any finite group G we give bounds on the order and exponent of Yc+l H. For G equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as -yc+lH. Our proofs involve: (i) the Baer invariants of G, (ii) the Schur multiplier M(L, G) of G relative to a normal subgroup L, and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.

Nilpotent actions on non-abelian tensor products of groups

Nilpotent actions on non-abelian tensor products of groups

On the nilpotency class and solvability length of nonabelian tensor products of groups

The nonabelian tensor product $G \otimes H$ is defined for a pair of groups G and H, provided G and H act on each other in a compatible fashion. The goal of this paper is to give bounds on the nilpotency class and solvability length of $G \otimes H$ , provided such information is given in context with G and H. The bounds are given in terms of D H (G), the derivative subgroup of G afforded by the action of H on G, and D G (H), the analogous subgroup of H. These derivative subgroups reduce to the commutator subgroup of G if G = H with all actions conjugation. We conclude with an example showing that, in contrast to the cases of the tensor square, the tensor product of a pair of groups of nilpotency class two is not necessarily abelian.

Nonabelian tensor products and nonabelian homology of groups

The non-abelian tensor product of groups introduced by Brown and Loday is generalized. The homology groups of groups are constructed with coefficients in any group, as the left derived functors of the non-abelian tensor product, which generalize the classical theory of homology of groups. Exact sequences of the non-abelian homology groups and their application to algebraic K-theory of noncommutative local rings are given.

The non-abelian tensor product of groups and related constructions

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.

Some computations of non-abelian tensor products of groups

On considere le produit tensoriel generalise G⊗H des groupes G, H. On presente le calcul de G×G