Graph Of Finite Group Research Articles

On the Gorenstein cohomological dimension of group extensions

The Gorenstein cohomological dimension is defined for any group G and coincides with the virtual cohomological dimension vcdG, whenever the latter is defined and finite. Unlike the virtual cohomological dimension, the Gorenstein cohomological dimension behaves well with respect to extensions, finite graphs of groups and ascending unions. In this paper, we study the Gorenstein cohomological dimension GcdkG of groups G, which are of type FP∞ over a commutative ring k. We show that if 1⟶N⟶G⟶Q⟶1 is an extension of groups with N of type FP∞ over a field F and GcdFQ<∞, then GcdFG=GcdFN+GcdFQ. We also show that for any group G of type FP∞ over Z with GcdZG<∞, there exists a field F such that GcdFG=GcdZG. This implies, in particular, that if G is a group of type FP∞ over Z and G is an extension of N by itself, then GcdZG=2GcdZN.

Intersection Normal Graphs of Finite Groups

Intersection Normal Graphs of Finite Groups

Equal‐Square Graphs Associated with Finite Groups

The graphical representation of finite groups is studied in this paper. For each finite group, a simple graph is associated for which the vertex set contains elements of group such that two distinct vertices x and y are adjacent iff x2 = y2. We call this graph an equal‐square graph of the finite group G, symbolized by ES(G). Some interesting properties of ES(G) are studied. Moreover, examples of equal‐square graphs of finite cyclic groups, groups of plane symmetries of regular polygons, group of units U(n), and the finite abelian groups are constructed.

Subgroup Graphs of Finite Groups

Let G be a fnite group with the set of subgroups of G denoted by S(G), then the subgroup graphs of G denoted by T(G) is a graph which set of vertices is S(G) such that two vertices H, K in S(G) (H not equal to K)are adjacent if either H is a subgroup of K or K is a subgroup of H. In this paper, we introduce the Subgroup graphs T associated with G. We investigate some algebraic properties and combinatorial structures of Subgroup graph T(G) and obtain that the subgroup graph T(G) of G is never bipartite. Further, we show isomorphism and homomorphism of the Subgroup graphs of finite groups. Let be a finite group with the set of subgroups of denoted by , then the subgroup graphs of denoted by is a graph which set of vertices is such that two vertices , are adjacent if either is a subgroup of or is a subgroup of . In this paper, we introduce the Subgroup graphs associated with . We investigate some algebraic properties and combinatorial structures of Subgroup graph and obtain that the subgroup graph of is never bipartite. Further, we show isomorphism and homomorphism of the Subgroup graphs of finite groups.

Graceful labeling of power graph of group Z2k−1 × Z4

The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let [Formula: see text] be a finite abelian 2-group of order [Formula: see text] where [Formula: see text]. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on [Formula: see text].

A note on finiteness properties of graphs of groups

We show that if G G is of type F n \mathcal {F}_n , and G G splits as a finite graph of groups, then the vertex groups are of type F n \mathcal {F}_n if the edge groups are of type F n \mathcal {F}_n .

On the enhanced power graph of a finite group

The enhanced power graph of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number, and matching number of enhanced power graphs of finite groups. We first study these graph invariants for when G is any finite group and then determine them when G is a finite abelian p-group, the dihedral group or the semidihedral group If G is any of these groups, we prove that is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of for any finite abelian group G. These results along with certain known equalities yield the edge connectivity, vertex covering number, and edge covering number of enhanced power graphs of the respective groups as well.

Dynamics of monad graph of finite group

Monad graph was introduced by V.I. Arnold in 2003 and many result were investigated in . In this paper is we study the properties of monad graph generated by dynamical system defined on finit group which is isomorphic to Cn cyclic group of order n and Dihedral group D 2n .

Proper connection of power graphs of finite groups

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

Quiver Representations, Group Characters, and Prime Graphs of Finite Groups

We develop a general theory of quiver representations $\mathcal{F}$ which is motivated by investigating group representations in different characteristics simultaneously. A feature of $\mathcal{F}$ is to assign to each vertex $a$ of the quiver a finite set $\mathcal{F}_{a}$. This result provides a uniform explanation of many objects in group character theory. Furthermore, we give a characterization of when the prime graph $\Gamma(G)$ of a finite group $G$ is disconnected by using group characters. And then, this result on $\Gamma(G)$ can be rephrased in terms of certain quiver representations.

Combinatorial properties of power graphs of finite groups of prime order elements

For a finite group G, the power graph P(G) represents the graph with elements of G as and if and only if one of u or v is a power of the other. In this paper, some combinatorial properties of power graph of a group G having all nonidentity elements of prime order are investigated. The necessary and sufficient conditions for P(G) to be power graph of such group are obtained. Further, unicylic, bicyclic and cacti power graphs of finite groups are characterized. Moreover, the eigenspectra of power graphs of these groups are also computed.

Residually finite tubular groups

A tubular group G is a finite graph of groups with ℤ2 vertex groups and ℤ edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.

Some Metrical Properties of Lattice Graphs of Finite Groups

This paper is concerned with the combinatorial facts of the lattice graphs of Z p 1 × p 2 × ⋯ × p m , Z p 1 m 1 × p 2 m 2 , and Z p 1 m 1 × p 2 m 2 × p 3 1 . We show that the lattice graph of Z p 1 × p 2 × ⋯ × p m is realizable as a convex polytope. We also show that the diameter of the lattice graph of Z p 1 m 1 × p 2 m 2 × ⋯ × p r m r is ∑ i = 1 r m i and its girth is 4.

Order Divisor Graphs of Finite Groups Un and K4

Order Divisor Graphs of Finite Groups <i>Un and K4</i>

Eulerian properties of non-commuting and non-cyclic graphs of finite groups

ABSTRACTFor a non-abelian group G, the non-commuting graph Γ(G) has G−Z(G) as its vertex set and two vertices x and y are connected by an edge if xy≠yx. For a non-cyclic group G, the non-cyclic graph has G−Cyc(G) as its vertex set, where Cyc(G) = {x|⟨x,y⟩ is cyclic, for all y∈G} and two vertices x and y are connected by an edge if ⟨x,y⟩ is not cyclic. We show that for all finite non-abelian groups G, Γ(G) is eulerian if and only if is eulerian. We investigate the eulerian properties of these graphs for various G showing, in particular, that Γ(G) (and hence ) is path-eulerian if and only if G≅S3.

Non-exterior square graph of finite group

We define the non-exterior square graph ??G which is a graph associated to a non-cyclic finite group with the vertex set G\?Z(G), where ?Z(G) denotes the exterior centre of G, and two vertices x and y are joined whenever x ^ y ? 1, where ^ denotes the operator of non-abelian exterior square. In this paper, we investigate how the group structure can be affected by the planarity, completeness and regularity of this graph.

АППРОКСИМИРУЕМОСТЬ ФУНДАМЕНТАЛЬНОЙ ГРУППЫ КОНЕЧНОГО ГРАФА ГРУПП КОРНЕВЫМ КЛАССОМ ГРУПП

Let \(\mathcal{K}\) be an abstract class of groups. Suppose \(\mathcal{K}\) contains at least a non trivial group. Then \(\mathcal{K}\) is called a root-class if the following conditions are satisfied: 1. If \(A \in \mathcal{K}\) and \(B \leq A\), then \(B \in \mathcal{K}\). 2. If \(A \in \mathcal{K}\) and \(B \in \mathcal{K}\), then \(A\times B \in \mathcal{K}\). 3. If \(1\leq C \leq B \leq A\) is a subnormal sequence and \(A/B, B/C \in \mathcal{K}\), then there exists a normal subgroup \(D\) in group \(A\) such that \(D \leq C\) and \(A/D \in \mathcal{K}\). Group \(G\) is root-class residual (or \(\mathcal{K}\)-residual), for a root-class \(\mathcal{K}\) if, for every \(1 \not = g \in G\), exists a homomorphism \(\varphi \) of group \(G\) onto a group of root-class \(\mathcal{K}\) such that \(g\varphi \not = 1\). Equivalently, group \(G\) is \(\mathcal{K}\)-residual if, for every \(1 \not = g \in G\), there exists a normal subgroup \(N\) of \(G\) such that \(G/N \in \mathcal{K}\) and \(g \not \in N\). The most investigated residual properties of groups are finite groups residuality (residual finiteness), \(p\)-finite groups residuality and soluble groups residuality. All there three classes of groups are root-classes. Therefore results about root-class residuality have safficiently enough general character. Let \(\mathcal{K}\) be a root-class of finite groups. And let \(G\) be a fundamental group of a finite graph of groups with finite edges groups. The necessary and sufficient condition of virtual \(\mathcal{K}\)-residuality for the group \(G\) is obtained.

Quantum decomposition of random walk on Cayley graph of finite group

In the paper, A quantum decomposition (QD, for short) of random walk on Cayley graph of finite group is introduced, which contains two cases. One is QD of quantum random walk operator (QRWO, for short), another is QD of Quantum random walk state (QRWS, for short). Using these findings, I finally obtain some applications for quantum random walk (QRW, for short), which are of interest in the study of QRW, highlighting the role played by QRWO and QRWS.

On the Genus of the Nilpotent Graphs of Finite Groups

The nilpotent graph of a group G is a simple graph whose vertex set is G∖nil(G), where nil(G) = {y ∈ G | ⟨ x, y ⟩ is nilpotent ∀ x ∈ G}, and two distinct vertices x and y are adjacent if ⟨ x, y ⟩ is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.

On Laplacian spectrum of power graphs of finite cyclic and dihedral groups

The power graph of a finite group G is the graph whose vertices are the elements of G and two distinct vertices are adjacent if and only if one is an integral power of the other. In this paper, we study Laplacian spectrum of the power graph of additive cyclic group and the dihedral group . We show that the Laplacian spectrum of is the union of that of and . We find algebraic connectivity of and give bounds of the same for .