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.

In the same vein, 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. Many years after Euler's research work on the bridges of Konigsberg city, Cayley [6] used the generators of a finite group to define a graphical structure called the Cayley graph of finite group , he further showed that every group of order can be represented by a strongly connected diagraph of n vertices [7]. Afterwards, in the last few decades, his view of diagraph has since been extended to different and modified graph of algebraic structures. Hence, more algebraic studies through the properties of these modified graphs have become topics of interest to many around the globe (See [8], [9], [10], [11], [12], [13], [14], [15]).
This study, the subgroup graph of finite groups like [8], [9], [10], [11], [12], [13], [14], [15], will focus on finite groups , however, the choice of it vertex set is the subgroups of . In the literature, vertex set of graphs of finite groups are always the elements n G, a deviation from this norm is the motivation for this study.
1.1. Preliminaries. We state some known and useful results which will be needed in the proof of our main results and understanding of this paper. For the definitions of the basic terms and results given in this section ( [16], [17], [18], [19], [20], [21], [22], [23]

Research Methodology
This article is not a variable base research, however, well known algebraic definitions and results were used to investigate the algebraic and combinatorial properties of the subgroup graph of finite groups.

Results and Discussion
The Subgroup graphs of finite groups is introduce in this section. We begin with the definition and notion of the Subgroups graph of a finite group. Below, we give an example of Subgroups graph.    If both and are normal in , then is also a vertex on .

2.
If alone is normal in , then is a adjacent to vertex on .

3.
If is normal in , then and is also a vertex on .

4.
If both and are normal in , then is also a vetex on .
Proof: From Theorem 1.20, the results follows.
Theorem 3.9. Let be a finite group of non prime order , then the subgroup graph of is never a star graph.
Proof: Suppose on the contrary, let the subgroup graph of a finite group of non prime order be a star graph; then it implies that all other are subgroup to only an arbitrary subgroup , but by Remark 3.2(2), every group has two trivial subgroups which are adjacent to all other . So, the graph can not be a star graph, since, there is more than one vertex that is adjacent to all the vertices of .    Proof: Suppose and are non nilpotent finite groups such that there is an isomorphic map between and , then we can safely say there is also isomorphic map between the commutator subgroups of and , which shows the isomorphic relationship between and . Also, since the commutator subgroup of is adjacent to on then the commutator subgroup of is also adjacent to on .

Conclusion
This study has highlighted some algebraic properties and combinatorial structures of Subgroups graph of finite groups. The connections between the Subgroups graphs of finite groups upto homomorphism and isomorphism were also studied and further looked at the relationships between the subgroups graphs of symmetric groups, , dihedral groups and the alternating groups An, when .