Generalization of the non-commuting graph of a group via a normal subgroup

Kakeri, F.; Erfanian, Ahmad; Mansoori, Farzaneh

doi:10.2306/scienceasia1513-1874.2016.42.231

Cited by 3 publications

(3 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, the order graph was defined by Bilal and Ahmad [4] as a directed graph whose vertices are the elements of the group order classes. Next, the subgroup graph was formally defined in 2015 by Kakeri and Erfanian [5]. The subgroup graph is a simple directed graph associated with subgroups of a finite group and this graph was extended to a non-normal subgroup graph defined by Nabilah et al [6].…”

Section: Introductionmentioning

confidence: 99%

The Application of GAP Software in Constructing the Non-Normal Subgroup Graphs of Alternating Groups

Rahin

Sarmin

Ilangovan

2022

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

A graph in group theory is constructed by using any elements of a group as a set of vertices. Some of the properties of a group are used to form the edges of the graph. A finite group can be represented in a graph by its subgroup structure. A subgroup H of a group G is a subset of G, where H itself is a group under the same operation as in G, whereas a subgroup H is said to be a normal subgroup if its left and right cosets coincide. The non-normal subgroup graph of a group G is defined as a directed graph with a vertex set G and two distinct elements x and y are adjacent if xy ∈ H. In this paper, the non-normal subgroups graph of alternating groups for order twelve is determined by using GAP software. The graphs are found to be a union of complete digraphs and directed graphs with the same pattern depending on the order of the non-normal subgroups.

show abstract

Section: Introductionmentioning

confidence: 99%

The Application of GAP Software in Constructing the Non-Normal Subgroup Graphs of Alternating Groups

Rahin

Sarmin

Ilangovan

2022

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

show abstract

“…[2] introduced the definition of non-commuting graph of a group, see also [3,4,5,6,7]. As generalization of non-commuting graph of a group, Erfanian, et.al [8] introduced the definition of generalization of the non-commuting graph of a group via a normal subgroup. Erfanian, et.…”

Section: Introductionmentioning

confidence: 99%

Generalized Non-Braid Graphs of Rings

Cahyati,

Maharani,

Nurhayati

et al. 2022

JFMA

View full text Add to dashboard Cite

In this paper, we introduce the definition of generalized non-braid graph of a given ring. Let $R$ be a ring and let $k$ be a natural number. By generalized braider of $R$ we mean the set $B^k(R):=\{x \in R~|~\forall y \in R,~ (xyx)^k = (yxy)^k\}$. The generalized non-braid graph of $R$, denoted by $G_k(\Upsilon_R)$, is a simple undirected graph with vertex set $R\backslash B^k(R)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $(xyx)^k \neq (yxy)^k$. In particular, we investigate some properties of generalized non-braid graph $G_k(\Upsilon_{\mathbb{Z}_n})$ of the ring $\mathbb{Z}_n$.

show abstract

“…The generalization of non-commuting graph of a group was given by Kakeri et al . in [6], where they investigated the connectivity, regularity and planarity of the graph and concurrently, give the numerical invariants of the graph which happen to be the improvement of the result given for non-commuting graphs. Abd Rani et al .…”

Section: Introductionmentioning

confidence: 99%

A Systematic Approach to Group Properties Using its Geometric Structure

Bello

Ali

Zulkifli

2020

Eur. J. Pure Appl. Math.

View full text Add to dashboard Cite

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

show abstract

Generalization of the non-commuting graph of a group via a normal subgroup

Cited by 3 publications

References 1 publication

The Application of GAP Software in Constructing the Non-Normal Subgroup Graphs of Alternating Groups

The Application of GAP Software in Constructing the Non-Normal Subgroup Graphs of Alternating Groups

Generalized Non-Braid Graphs of Rings

A Systematic Approach to Group Properties Using its Geometric Structure

Contact Info

Product

Resources

About