Subgroup Graphs of Finite Groups

Ojonugwa Ejima,Department Of Mathematics, Usmanu Danfodiyo University ,Abor Isa Garba,Kazeem Olalekan Aremu

doi:10.24071/ijasst.v3i2.3765

Ojonugwa Ejima, Department Of Mathematics, Usmanu Danfodiyo University + Show 2 more

Open Access

https://doi.org/10.24071/ijasst.v3i2.3765

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Abstract

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.

Highlights

One of the mathematical tools for studying symmetries of object is group theory, several structures in the field of algebra are depicted through groups
According to [1], the rebirth of the axiomatic method and the view of mathematics as a human activity in the nineteenth century forms the major development that change the bearing on the evolution of group theory as a mathematical concept. [1], further noted that the evolution had caused the previous classical algebra polynomial equations
In the last two decades, many studies have related graphs to group theory, providing a more easier way to visualize the concept of group; this relation brings together two important branches of mathematics, and has opened up a new wave of research with a better understanding of the fields

Summary

Introduction

One of the mathematical tools for studying symmetries of object is group theory, several structures in the field of algebra are depicted through groups. This mathematical concept has evolved rapidly since it discovery in the sixteenth century. Vertex set of graphs of finite groups are always the elements n G, a deviation from this norm is the motivation for this study. [20] A walk in a connected graph that visits every vertex of the graph exactly once without repeating the edges is called Hamiltonian path. Definition 1.9. [21] A group consists of a set with a binary operation * on satisfying the following four conditions: p-ISSN 2655-8564, e-ISSN 2685-9432

Inverse

There exists with

Results and Discussion

The has a diameter and radius of if

Conclusion