International Journal of Applied Sciences and Smart Technologies International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 225 Subgroup Graphs of Finite Groups Ojonugwa Ejima 1,* , Abor Isa Garba 1 , Kazeem Olalekan Aremu 1 1 Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria * Corresponding Author: unusoj1@yahoo.com (Received 08-10-2021; Revised 29-12-2021; Accepted 29-12-2021) Abstract 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. Keywords: subgroup, graph, finite group 1 Introduction One of the mathematical tools for studying symmetries of object is group theory, hence, several structures in the field of algebra are depicted through groups. This mathematical concept has evolved rapidly since it discovery in the sixteenth century. 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 International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 226 transited to the modern algebra of axiomatic systems of the nineteenth century. Meanwhile, this concept has been applied in the field of physics, chemistry and biology, (see [2], [3], [4], [5]) for details. 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]). A graph is a combinatorial structure formed by finite non-empty set where is the set of vertices viewed as points and is the set of edges viewed as line joining the points. The cardinality of is called the order of while the cardinality of is called the size of . The degree of a vertex in a graph denoted by is the number of edges incident to it, that is the number of edges connecting . A graph is said to have parallel edges if there are more than one edges which join the same pair of International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 227 distinct vertices. A loop on the other hand is an edge that joins a vertex to itself while a walk of length in a graph with vertex set consist of an alternating sequence of vertices and edges consecutive elements of which are incident, that begins and ends with a vertex. Definition 1.1. [20] A complete graph is a simple undirected graph in which has at least one vertex and every arbitrary pair of distinct vertices is joint by a unique edge. while a connected graph on the other hand is a graph where there is an edge between every pair of vertices. Remark 1.2. Note that every complete graph is necessary connected but connected graphs are not necessary complete. Definition 1.3. [20] A walk in a connected graph that visits every vertex of the graph exactly once without repeating the edges is called Hamiltonian path. If this walk starts and ends at the same vertex, the walk is called a Hamiltonian circuit or cycle. Remark 1.4. A graph that contains a Hamiltonian cycle is said to be Hamiltonian. Theorem 1.5. (Sylow’s First Theorem) [21] Let be a finite group of order , where is a prime, and are positive integers and . Then has a subgroup of order for all satisfying . Definition 1.6. [20] In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i.e. every vertex has the same degree or valency; a regular graph can be an x-regular graph where every vertex of the graph have the same degree . Definition 1.7. [20] The distance between two vertices is the length of the shortest path between and and it’s denoted by Eccentricity of a vertex in a graph is define as { }. The minimum and maximum eccentricity in a graph are called radius, rad and diameter, diam( ) of the graph respectively. Lemma 1.8. [20] Let G and G′ be any two finite groups. If then and But the converse is not true. Definition 1.9. [21] A group consists of a set with a binary operation * on satisfying the following four conditions: International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 228 1. Closure: , we have . 2. Associativity: , we have . 3. Identity: There is an element satisfying for all . 4. Inverse: For all , there is an element satisfying . (where is as in the Identity Law) Definition 1.10. [20] A finite group is a group containing finite number of elements. The order of a finite group is the number of elements in . Definition 1.11. [21] A subset of a group is called a subgroup if it forms a group in its own right with respect to the same operation on . Definition 1.12. [21] Let and be groups. A homomorphism from to is a map which preserves the group operation. Definition 1.13. [21] A subgroup of is said to be a normal subgroup if it is the kernel of a homomorphism. Equivalently, is a normal subgroup if its left and right cosets coincide: for all . We write ” is a normal subgroup of ” as , we write . If is a normal subgroup of , we denote the set of (left or right) cosets by . We define an operation on by the rule for all Definition 1.14. [24] Let and be elements of a group such that yields an element of and is defined by , the collections of arbitrary in forms the commutator subgroup of . Lemma 1.15. [16] Suppose and e is positive integer. Then 1. . 2. . 3. . 4. . Lemma 1.16. [25] Let be a group with subgroup of and subgroup of Set and the following holds 1. , and if is abelian, then 2. If is abelian, then the minimal number of elements needed to generate and , then . International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 229 3. If is abelian and the minimal number of elements needed to generate , there exists such that . 4. If is cyclic, then is cyclic Lemma 1.17. [25] Suppose 〈 〉 is a finite group with ′ abelian and of order . If is the commutator subgroup of , then { } Theorem 1.18. [25] Let be group with abelian and 1. There exists with In particular, if , then can be taken to be any element of . 2. Let . If either then 3. is equal to the commutator subgroup of . Lemma 1.19. [25] Let be a subgroup of , element of and . If is abelian, then . Theorem 1.20. [26] Let be a group and let be two subgroups of and define , then 1. If both and are normal in , then is also a normal subgroup of . 2. If alone is normal in , then is a normal subgroup of . 3. If is normal in then and is a normal subgroup of . 4. If both and are normal in , then is a normal subgroup of . Theorem 1.21. (Sylow’s theorems) Let be a group of order , where is a prime, , and does not divide . Then: 1. that is subgroups exist. 2. All subgroups are conjugate in . 3. Any subgroup of is contained in a subgroup. 4. . Lemma 1.22. (Lagrange’s theorem) [22] Let be a subgroup of a finite group . Then the order of divides the order of . International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 230 2 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. 3 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. Definition 3.1. Let be a finite group and be the set of subgroups of . Then the Subgroup graph of is the graph with vertex set and the edge set {{ } } Remark 3.2. Let be a finite group of order , some clear consequences of the definition of subgroup graphs of finite groups are 1. The subgroup graph is a simple graphs, thus, there are no loops nor multiple edges. 2. The trivial subgroups of are adjacent to every other vertices on 3. Since the trivial subgroups of are adjacent to every other vertices on then the graph is connected. 4. The has a diameter and radius of if . Below, we give an example of Subgroups graph. Example 3.3. Let the group of integer modulo 6 under addition then the following is the undirected Subgroups graph of See Figure 1. Figure 1. Subgroups graph of . International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 231 Theorem 3.4. [20] A simple graph is bipartite if and only if it does not have any odd cycle. Remark 3.5. Let be a finite group, the subgroup graph of is never bipartite. Theorem 3.6. Let be a finite group of prime order and let be the set of subgroups of , then the subgroups graph of is a straight line with only two vertices. Proof: Let be a finite group of prime order and let be the set of subgroups of , then by Lemma 1.22, the order of every divides the order of but the order of is a prime which can only be divided by itself of 1. Thus the only subgroups of this group are {{ } }; and they are adjacent. Remark 3.7. Let be a finite group and let be the subgroups graph of . Then the vertex set and edge set therefore, the subgroups graph of a finite group can never be empty. Theorem 3.8. Let be a group and let , be two subgroups of and let be the subgroup graph of . Define { }, then 1. 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 . International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 232 Theorem 3.10. Let and be two isomorphic finite groups. Then the subgroup graph of is isomorphic to subgroup graph of . Proof: Suppose and are two isomorphic finite groups, then from Lemma 1.8, . Thus, . Theorem 3.11. Let be a group of order , where is a prime, and does not divide . Suppose is the subgroup graph of , and are the vertex and edge sets of respectively. Then: 1. There must exist Sylow -subgroups of which are conjugate in and are adjacent to each other on the subgroup graph of . 2. There is a vertex on the subgroup graph of that is adjacent to a vertex that is a Sylow -subgroup of . Proof: 1. To show that there must exist Sylow -subgroups of which are conjugate in and are adjacent to each other on the subgroup graph of , then it suffices if we can show to be conjugate and subsequently establish an edge between and . From Theorem 1.21, it is established that the group has some Sylow -subgroups. So, let be a Sylow -subgroup of , and let be the set of all distinct conjugates of . Suppose is the order , we need to establish that cannot divide . Since each of the is a conjugate to , it implies every element of is in the orbit of . So using the formula for orbit size (where is the normalizer of ). However, Lagrange’s theorem established that and clearly, is a subgroup of and it contains as a factor and a maximum power can assume. Thus, and contains no factor of and so does not divide . 2. Suppose is any -subgroup of , it will suffice if we can show . Let act on { }, by conjugation. Clearly, the the orbits of this action will partition . Suppose the distinct orbits are the orbits of { } then the orbit of . To compute the orbit for International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 233 any , , since is the stabilizer of under the action of . Then the size of each orbit has to divide , which is a power of . Though doesn’t divide so there is no dividing all the terms and else would divide their sum and also . Assume which means and since is a −subgroup . This implies so every element of is also in then , therefore, the −subgroup is a subgroup of the arbitrary Sylow −subgroup of . So, and are adjacent of the subgroup graph of . Theorem 3.12. Let and be two finite groups and be a group homomorphism. Suppose there is an , a normal subgroup of and an , a normal subgroup of such that is adjacent to on the subgroup graph of and is adjacent to on the subgroup graph of . Then 1. is adjacent to on the subgroup graph of . 2. is adjacent to on the subgroup graph of . Proof: Suppose and are two finite groups and is a group homomorphism, if there is an , a normal subgroup of and an , a normal subgroup of such that is adjacent to on and is adjacent to on then to show that is also adjacent to on and is adjacent to on ; it will suffice if we can show to be a normal subgroup of and to be a normal subgroup of respectively. 1. Let , since is a group homomorphism and is normal in . Thus, is normal in 2. Let a be an arbitrary element of , then the set satisfies that since is normal in . Thus, for every . This shows that is a normal subgroup of . International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 234 Corollary 3.13. [27],[28] The alternating group is a subgroup of the symmetric group . Theorem 3.14. Let , and be the subgroups graphs of symmetric groups, , dihedral groups and the alternating groups , . Suppose and are vertices on and respectively, then both and are also vertices on and are adjacent to if and only if is adjacent to on and is adjacent to on Proof: Let , and be the subgroups graphs of symmetric groups, , dihedral groups and the alternating groups , . To show that and which are vertices on and respectively are also vertices on and are adjacent to if and only if is adjacent to on and is adjacent to on , then it suffices, if we can show both and to be subgroups of . Assume that is a subgroup of and is a subgroup of . Then observe the structure of the group of symmetries of a regular −gon in a plane (dihedral group ( )), it is isomorphic to a subgroup of then it is a proper subgroup of . Also, by Corollary 3.13, is a subgroup of . Moreover, since and by implication, they are also subgroups of and hence are vertices on Conversely, assume that is adjacent to on and is adjacent to on , then by Definition 3.1, and . But both and are subgroups of , also, by implication, they are adjacent to . Theorem 3.15. [29] If , then the number of subgroups of the dihederal group is . Where is the number of divisors of and is the sum of divisors of . Remark 3.16. Let be a dihedral group of order , then the number of vertices on the subgroups graphs of the dihederal group is , where is the number of divisors of and is the sum of divisors of . International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 235 Theorem 3.17. Let be a commutator subgroup of a finite group of order , suppose there exist a normal subgroup of such that is Abelian, then and are adjacent on the subgroup graphs of . Proof: Let be a finite group of order and the commutator subgroup of . If there exist a normal subgroup of such that is Abelian, then to show that and are adjacent on (the subgroup graph of ), we must show that either or . Note that is normal in and is Abelian, then for , we have and using the definition of Coset multiplication . Which implies , where . Similarly, and since and are arbitrary then any commutator in is an element of and since is a subgroup of then any finite product of commutators in is an element of and thus . Theorem 3.18. Let be a commutator subgroup of a finite group of order and let be a subgroup of . If there exist an such that , then and are adjacent on the subgroup graphs of . Proof: By Lemma 1.15(2) and using the method of [25], the map sending to is a homomorphism. Thus, the image map of is the subgroup Theorem 3.19. (Schur Zassenhaus) [16] Let be a finite group and write where . If has a normal subgroup of order then it has a subgroup of order . Remark 3.20. Let be a finite group and write where . Then the vertex set of the subgroup graph of contains at least two vertices which orders are a and b respectively. International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 236 Theorem 3.21. Let be a finite nilpotent group, such that is an abelian p-group with the minimal number of elements needed to generate , then is a vertex on the subgroup graph and it is adjacent to . Proof: Suppose is a finite nilpotent group, such that is an abelian p-group with , then to show that is a vertex adjacent to on the subgroup graph of , it will suffice if we can show to be equal to , the commutator subgroup of . Now, since is finite, we assume an arbitrary such that and obviously, is normal in and by Theorem 3.19 (Schur Zassenhaus theorem) and the methods in [25], where { }. Also, by Lemma 1.16(3), and Lemma 1.17, we set and if there exist such that and { } . Thus, we can assume , and they also exists with Since is nilpotent, by Theorem 1.18 and Lemma 1.19, (⋃ ) ⋃ Lemma 3.22. Let and be two non nilpotent finite groups, such that there is an isomorphism of and , if the commutator subgroup of is adjacent to on the subgroups graph of , then the commutator subgroup of is also adjacent to on the subgroups graph 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 . Example 3.23. Let be a quaternion group generated by the following matrices ( ) ( ) ( ) ( ) International Journal of Applied Sciences and Smart Technologies Volume 3, Issue 2, pages 225–240 p-ISSN 2655-8564, e-ISSN 2685-9432 237 [30], using the matrix multiplication obtained { }, observe that the subgroups of consists of itself and of the cyclic subgroups 〈 〉 { } 〈 〉 { } 〈 〉 { } 〈 〉 { } 〈 〉 { } and the following is the subgroups graph of . See Figure 2. Figure 2. Subgroups graph of Q8. 4 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 . Acknowledgements The authors wish to thank the reviewers for their helpful comments and recommendations on this article. 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