The Non-Abelian Tensor Square Graph Associated to a Symmetric Group and its Perfect Code

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.

Non-cancellation and non-abelian tensor squares

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.

The study of graph properties has gathered many attentions in the past years. The graph properties that are commonly studied include the chromatic number, the clique number and the domination number of a finite graph. In this study, a type of graph properties, which is the perfect code is studied. The perfect code is originally used in coding theory, then extended to other fields including graph theory. Hence, in this paper, the perfect code is determined for the commuting zero divisor graphs of some finite rings of matrices. First, the commuting zero divisor graph of the finite rings of matrices is constructed where its vertices are all zero divisors of the ring and two distinct vertices, say x and y, are adjacent if and only if xy = yx = 0. Then, from the vertex set of the graph, the neighborhood elements of the vertices are determined in order to compute the perfect codes of the graph.

Non-abelian tensor product and circular orderability of groups

reduction graphs be ARG. It turns out that there are graphs in ARG that are not (isomorphic to) reduction graphs. To obtain a characterization we need one more property of reduction graphs: the ability to linearly order the vertices to resemble its (in general not unique) underlying legal string, as done in Fig. 11. To make this linear order of vertices explicit, we introduce a third set of edges, called merge edges , to the reduction graph as done in Fig. 14. s 2 2 7 7 4 4 7 7 3 3 5 5 3 3 4 4 2 2 6 6 5 5 6 6 t Fig. 14. Merge edges are added to the reduction graph of Fig. 11. Now, when is a set of edges M for G ∈ ARG a set of merge edges? Like desire edges, they have the properties that (1) the edges connect vertices with a common label and (2) each vertex except s and t is connected to exactly one merge edge. Moreover, M and the set E2 are disjoint – no desire edge is parallel to a merge edge. Finally, the reality edges and merge edges must allow for a path from s to t passing each vertex once. This last requirement is equivalent to the fact that the reality and merge edges induce a connected graph. If it is possible to add a set of merge edges to the graph, then it is not difficult to see that the graph is isomorphic to a reduction graph Ru. Indeed, we can identify such a u for this reduction graph by simply considering the alternating path from s to t over the reality and merge edges. The orientation (positiveness or negativeness) of each pointer is determined by the crossing or non crossing of the desire edges (exactly as we defined the notion of reduction graph). To characterize reduction graphs we need the notion of a pointer-component graph. Given an abstract reduction graph, a pointer-component graph describes how the labels of that abstract reduction graph are distributed among its connected components. Definition 5.14. Let G ∈ ARG. The pointer-component graph of G, denoted by PCG, is a multigraph (ζ, E, e), where ζ is the set of connected components of G, E = dom(G) and e is, for e ∈ E, defined by e(e) = {C ∈ ζ | C contains vertices labelled by e}. The pointer-component graph of G = Ru of Fig. 12 is given in Fig. 15. We have ζ = {C1, C2, C3, R} where R is the linear component and the other elements are cyclic components of Ru. It is shown in [7] that, surprisingly, G ∈ ARG has a set of merge edges precisely when the pointer-component graph PCG is a connected graph. In other words: Theorem 5.15 ([7]). An abstract reduction graph G is isomorphic to a reduction graph iff PCG is a connected graph. Computational nature of gene assembly in ciliates 21

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes