Non-cancellation and non-abelian tensor squares

A set of vertices and edges forms a graph. A graph can be associated with groups using the groups' properties for its vertices and edges. The set of vertices of the graph comprises the elements of the group, while the set of edges of the graph is the properties and requirements for the graph. A non-abelian tensor square graph of a group is defined when its vertex set represents the non-tensor centre elements' set of G. Then, two distinguished vertices are connected by an edge if and only if the non-abelian tensor square of these two elements is not equal to the identity of the non-abelian tensor square. This study investigates the non-abelian tensor square graph for the symmetric group of order six. In addition, some properties of this group's non-abelian tensor square graph are computed, including the diameter, the dominating number, and the chromatic number. The perfect code for the non-abelian tensor square graph for a symmetric group of order six is also found in this paper.

One of the homological functors of a group, is the nonabelian tensor square. It is important in the determination of the other homological functors of a group. In order to compute the nonabelian tensor square, we need to get its independent generators and its presentation. In this paper, we present the calculation of getting the presentation of the nonabelian tensor square of the group. The presentation is computed based on its independent generators by using the polycyclic method.

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

In the theories of groups and Lie algebras, investigations of the properties of the non-abelian tensor product and their relations to the second homology groups are worthwhile. It is the purpose of the present paper to exhibit such investigations about the non-abelian tensor product of Leibniz algebras. The isomorphism between the non-abelian tensor square and non-abelian exterior square of a Lie algebra L, will enable us to set a simple connection between and . Furthermore, we shall relate the concepts of capability and solvability of a Leibniz algebra to its tensor square. Finally, we give an upper bound for the dimension of the non-abelian tensor square and the second homology of a nilpotent Leibniz algebra in terms of the dimension of its center and derived subalgebra.

We study the non-abelian tensor square modulo q of a group, where q is a non-negative integer, via an operator ν q in the class of groups. Structural properties and finiteness conditions of νq (G) are investigated. We compute the non-abelian tensor square modulo q of cyclic groups and develop a theory for computing νq (G) and some of its relevant sections for polycyclic groups G. This extends the existing theory from the case q = 0 to all non-negative integers q. Additionally, a table of examples is produced with the help of the GAP system.

We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of $$\mathfrak {sl}(m, n, D)$$ when $$m+n \ge 3$$ and D is a superdialgebra, solving, in particular, the problem when D is an associative algebra, superalgebra, or dialgebra. To accomplish this task, we use a different method than the standard studied in the literature. We introduce and use the non-abelian tensor square of Leibniz superalgebras and its relations with the universal central extension.

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

This article improves on an upper bound for the order of the non-abelian tensor square of a finite p-group G given in [3]. In particular, applying this for finite p-groups of order p n with factor group G/G′ of order p m , the bound p nm attains if and only if G is elementary abelian of rank n, quaternion group of order 8, or extra special p-group of order p 3 with odd exponent p.

Bieberbach groups are torsion free crystallographic groups. In this paper, our focus is on the Bieberbach groups with elementary abelian 2-group point group, The central subgroup of the nonabelian tensor square of a group is generated by for all in The purpose of this paper is to compute the central subgroups of the nonabelian tensor squares of two Bieberbach groups with elementary abelian 2-point group of dimension three.

An operator of weak commutativity between isomorphic groups, H and H ψ, was defined by Sidki as χ(H) = 〈H H ψ | [h,h ψ] = 1 for all h ∈ H〉, where ψ : h ↦ h ψ for all h ∈ H defines an isomorphism. It is known that the operator χ preserves group properties such as finiteness, solubility, and also nilpotency for finitely generated groups. We prove in this work that χ preserves the properties of being polycyclic or polycyclic-by-finite. As a consequence of this result, we conclude that the non-abelian tensor square H ⊗ H of a group H as defined by Brown and Loday preserves the property of being polycyclic-by-finite. This last result extends work of Blyth and Morse who proved that H ⊗ H is polycyclic if H is polycyclic.

Non-abelian tensor product and circular orderability 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 $G \otimes G$.