Investigation of Commuting Graphs for Elements of Order 3 in Certain Leech Lattice Groups

Assume that G is a finite group and X is a subset of G. The commuting graph is denoted by С(G,X) and has a set of vertices X with two distinct vertices x, y  X, being connected together on the condition of xy = yx. In this paper, we investigate the structure of Σ(G,X) when G is a particular type of Leech lattice groups, namely Higman–Sims group HS and Janko group J2, along with X as a G-conjugacy class of elements of order 3. We will pay particular attention to analyze the discs’ structure and determinate the diameters, girths, and clique number for these graphs.


Introduction and Preliminaries
It is believed that studying the action of a group on a graph is one of the best comprehensible ways of analyzing the structure of the group. Suppose that G is a group and X is a subset of G; the commuting graph denoted by С(G,X) has the set of vertices X with two vertices x, y  X, which are connected if x ≠y, where xy = yx. The commuting graphs were first illustrated by Fowler and Brauer in a seminal paper [1]. They were eminent for giving evidence of a prescribed isomorphism of an involution centralizer, where there is a limited number of non-abelian groups capable of containing it. These graphs were extremely vital for the works of the Margulis-Platanov conjecture [2], as the graphs mentioned in [1] have X = G\{1} where 1 is the identity element of G). When X is a conjugacy class of involution (conjugacy class X of G means that for any two elements x, y  X there are an element ISSN: 0067-2904 g G such that x g =y, whereas involution means that all elements of X have order 2), then the commuting graph is known as the commuting involution graph. Rowley, Hart (nѐe Perkins), Bates, and Bundy put their efforts into investigating the commuting involution graphs and supplying the diameters and disc sizes [3,4, 5 6]. Suppose that X is a conjugacy class of elements of order 3. Nawawi and Rowley [7] analyzed the С(G,X) when G is either a symmetric group S n or a sporadic group McL. The aim of this paper is to investigate the commuting graphs when G is a particular type of Leech lattice groups, such as the Higman-Sims group HS and Janko group J 2 , along with X as a Gconjugacy class for elements of order 3. The research involves scrutinizing the discs' structures and calculating the diameters, girths, and clique number for these graphs. From now, we shall assume that G is one of the aforementioned groups. Also, we let t be an element of order 3 in G and X = t G . It is clear that G, acting by conjugation, induces graph automorphisms of С(G,X) and is transitive on its vertices. For x  X and i  ℤ + , ∆ i (x) denotes the set of vertices of С(G,X) which has a distance i from x. Using the usual distance function for graphs, this distance function will be denoted by d(; ). We use G x (= C G (x)) to denote the stabilizer in G of x. Obviously, ∆ i (x) will be a union of certain Gxorbits. Therefore, we are looking for finding the Gx-orbits of X. In the computational group theory of Magma [8] and GAP [9], packages are considered most commonly utilized. In most steps of the algorithm, Gap will be dominant in the implementation, while the permutation character of the centralizer of t in G (C G (t)) may be verified using Magma and, hence, the number of C G (t)-orbits (Permutation Rank on X) under the action of X on C G (t) is calculated. Finally, we will use the online Atlas of Group Representations [10] to get a class name of the groups and we refer to it as The Online Atlas. For the aforesaid groups, the sizes of conjugacy classes for elements of order 3 and the permutation ranks on each class, as well as the structure of the centralizer of t in G, which can be seen in [11], are given in the next table.

Computational Method
Let x  ∆ i (t) and z  C G (t), then one can see immediately that x z  ∆ i (t). Thus, for a finite group G, each disc ∆ i (t) of the commuting graph Ϲ(G,X) is a union of specific C G (t)-orbits. The size of C G (t) -orbits under the action of conjugation on t G can be calculated by using the character table of the group, as we can see in the following result: Proposition 2.1. [12]. Let G be a group acting transitively on a finite set Ω, with a permutation character . Suppose that α  Ω and G α has exactly k orbits on Ω. Then ˂,˃ = k. The quantity k in Proposition 2.1 is called the permutation rank of G α on Ω .Therefore, the permutation rank of C G (t) on X is the number of C G (t)-orbits under the conjugation action on X. Now, let C be a G-conjugacy class. It is obvious that the set X C = {x  X : tx C} under the conjugation action of C G (t) breaks up into sub orbits. Thus, to find all the sub orbits of X, we have to identify the C G (t) -orbits of X C , for all those C, such that X C ≠ Ø . The next definition gives us the size of the set X C and, therefore, the size of sub orbits of X C : Definition 2.2. [13]. Let C i , C j and C k be the conjugacy classes of a finite group G. Then, for a fixed element g  C k , the following set is defined: a ijk = |{(g i , g j )  C i x C j | g i g j = g}|. Then, for all possible i, j, and k, the value a ijk is called a class structure constant for G. The next lemma will be used to compute the class structure constants for G. Lemma 2.3. [13]. Let G be a finite group with n conjugacy classes C 1 ,C 2 , …,C n . Then for all i, j, and k, we have g i as follows: where g i , g j , and g j are respectively in C i , C j and C k , and  is the irreducible character table in G.
For the proof of the above lemma, see [13], page 128, Lemma 2.12.. Now, since |X C | = |{(c, x)  C x X | cx = t}|= | a ijk |, then by employing Lemma 2.3, we get Therefore, from the complex character table of G, which is available in the GAP character table library, and using the GAP function of "Class Multiplication Coefficient", we immediately obtain the size of X C .

Diameters, Girths, and Clique Number
To determinate the diameters, girths, and clique number for the Ϲ(G,X), we will generated the graph by using the gap package YAGS [14] (specifically, Graph By Relation) in the next algorithm, which can be realized by using the definition of the commuting graph. The algorithm is given as follows: Algorithm 1 Input: The group G, t  G (the elements of order 3); i: X = t G the G-conjugacy class of t. ii: Relation = {the set of elements satisfies the condition : x ≠y and x*y = y *x} iii: С (G,X) is the graph generated by X and relation. v: Calculate Diameter(С(G,X)), Girth(С (G,X)) and Clique number (С (G,X)) . Output: The diameter, girth and the clique number of С (G;X).
The results in table 2 are obtained by applying Algorithm 1 on the groups specified in the table.
The last column of Table-2 represents the maximum elementary abelian group generated by the maximum clique.

Analyzing the Discs Structures
This section is dedicated to analyze the structures of the ∆i(t) of the commuting graph Ϲ(G, X).

C G (t)-Orbits of ∆ i (t)
The next algorithm is employed to break ∆ i (t) into C G (t)-sub orbits of X C , for all those С (G-Conjugacy class), such that X C ≠ φ, where their sizes are also provided.

Algorithm 2.
Input: the group G, t ∈ G, (the elements of order 3), С (G-Conjugacy Class) i: X → t G the G-conjugacy class of t ii: C G (t) → centralizer in G of t.
iii: O → the orbits in C G (t) of X. iv: |X C | =→ "Class Multiplication Coefficient" of C in X.
xx: Repeat the above steps and replace the Y i+1 with Y i and ∆ i+1 (t) with ∆ i (t) Output: The positions of the sets X C in the ∆ i (t) with their sizes.
For each graph in Table-2, we provide information about the discs structure. We should note that, in the next tables, the value n * m means the number and the size of C G (t)-orbits in certain ∆ i (t), respectively. The exceptional case is С(J 2 , 3A) as the graph is disconnected. The cases are: 1-Ϲ(J 2 , 3A): Form Table-1, the permutation rank of 3A is 10 and C G (t) C 3 . A 6 . In Table- Table-1, the permutation rank of 3B is 531. Form Table-2, the graph is connected with diameter 8. The centralizer of t ∈ 3B is isomorphic to C 3 x A 4 . The disc structure of the Ϲ(J 2 , 3B) is described in Table-3.

The subgroup < t, x >
For any x, y in the arbitrary C G (t)-orbit, the subgroup generated by t and x, i.e. <t,x>, is conjugated to the subgroup generated by t and y. This is obvious because if we conjugate one of them by t, we get the other. For each C G (t)-sub orbit of X C , the next algorithm provides the algebraic structure of the subgroup generated by t and random elements of C G (t)-sub orbits. We give the algebraic structure of each subgroup generated by < t, x > for a random element x in a C G (t)-suborbits of X C , with their numbers and sizes, for each connected graph mentioned in Tables-5  and 6. For more information about these subgroups, we refer to several previous works [11,13,15].
If Size Set C equal Size X C Stop vi: Else X C → X C \Set C vii: repeat the steps i to iv until the step v is true Output: The group algebraic structure of the subgroup generated by t and the suborbits of the sets X C in the ∆ i (t).
Azeez and Abd Aubad .  The collapsed adjacency matrix is an n × n matrix, such that n is the permutation rank given in the Table-1. The matrix is too big to fit in the paper. For this reason, we only give an example for applying Algorithm 4 to find the entry of the collapsed adjacency matrix. Now, we calculate some entries of the 399 × 399 collapsed adjacency matrix of Ϲ(HS, 3B). We provide the entries for such matrix since the technique we use is similar for any matrix size. Therefore, we just give 10 × 10 parts of 399 × 399 the collapsed adjacency matrix of Ϲ(HS, 3B), as showing below: