A4-Graph of Finite Simple Groups

Let G be a finite group and X be a conjugacy class of order 3 in G. In this paper, we introduce a new type of graphs, namely A4-graph of G, as a simple graph denoted by A4(G,X) which has X as a vertex set. Two vertices, x and y, are adjacent if and only if x≠y and x y -1 =y x -1 . General properties of the A4-graph as well as the structure of A4(G,X) when G D4(2) will be studied.


Introduction
Analyzing the group structures using graph structures, on which the group acts upon , can be an effective method which gives rise to many interesting results. Currently, this style of studying the algebraic properties of groups is the most common. There is a remarkable number of researches in this area, see for example [1,2,3]. Assume that G is a finite groups and X is a conjugacy class of order 3 in G. In this work, we present the A 4 -graph of G as a simple graph denoted by A 4 (G,X). The vertices set of A 4 -graph is X, and x,y  X are joined by an edge if and only if x≠y and x y -1 =y x -1 . Firstly, we note about the A 4 -graph, if x is adjacent to y, then the subgroup is generated by x and y, <x,y>A 4 . For this reason, we named the graph as A 4 -graph. Throughout this paper, we let G be a finite groups and X is a G-conjugacy class of order 3.. The aim of this work is to present the general properties of the A 4 -graph and describe certain features of A 4 (G,X), when G is an exceptional Lie type group of characteristic two 3 D 4 (2) information about this group, which can be found with details in [4]. For x  X, we define the i th disc of x,  i (x) , (i  N) to be  i (x)={y  X| d(x,y)=i}

ISSN: 0067-2904
Aubad Iraqi Journal of Science, 2021, Vol. 62, No. 1, pp: 289-294 290 where d( , ) is the usual distance metric on the graph A 4 (G,X). Certainly, G is acting by conjugation on X imbedding G in the group of graph automorphisms of A 4 (G,X). Obviously, G is a transitive on the vertices of A 4 (G,X). We now choose tX to be a fixed representative of the class X. The aim is to describe the disc structure of vertex t in A 4 (G,X). The diameter of A 4 (G,X) will be denoted by Diam A 4 (G,X) and is defined as For deep details about concepts of graph theorem, we may refer to [5]. Finally we shall rely upon the Atlas for the names of conjugacy classes of G [6].
1-General Properties of A 4 (G,X) Definition 1.1: Let G be a finite group. For G-conjugacy classes X of order 3, we assign a simple graph which is called A 4 -graph and denoted by A 4 (G,X), with vertices set being the set X, and two vertices x,y  X are adjacent if and only if x≠y and x y -1 =y x -1 .
The next examples are to illustrate the structure of A 4 -graph for certain finite groups. Examples 1.2 (1) Let GS 5 be a symmetric group of degree 5 and t= (3,4,5), then we have : (1,5,4) ]. The graph A 4 (G,X) is connected with Diam A 4 (G,X) =3. The disc structures of the graph A 4 (G,X) are: This can be achieved computationally by using the gap package YAGS [7] as we describe in the next procedure which proceeds as follows: Procedure 1 1. Define the group G and t. 2. Compute the G-Conjugacy classes X=t G . 3. Compute A 4 (G,X) by using the code GraphByRelation. 4. Draw the graph by using the code Draw. 5. Compute the diameter of the graph by using the code Diameter. 6. Set  0 (t)=t. For i in {1,2,3} Do 7. For y 1 in  i-1 (t) Do 8. For y 2 in X\  0 (t)   1 (t)…  i-1 (t) Do 9. If y 1 *y 2 -1 = y 2 *y 1 -1 Then 10. Add( i (t),y 2 ) Now, to simplify the graph drawing, we replace the vertex by its position in the set X . For example, we label 1 instead of the first elements in set X, which is (3,4,5), and 2 for the second element ( 3,5,4), and so on. (2) Dihedral group D 6 =< a,b | a 3 =b 2 =1, bab=a -1 > has only one class of elements of order 3. Then, the class consists of the set X={a,a -1 }. Clearly, a 2 ≠ a -2 , then the A 4 (D 6 ,X) is disconnected with single vertex connected components. Now, we will establish some general properties for A 4 (G,X).

Proposition 1.3: Let G be a finite group and X be a conjugacy class of order 3 in G. Then, graph
A 4 (G,X) has the following properties: Proof 1-From the definition of the graph A 4 (G,X), we have two vertices x,y  X are adjacent if they are distinct and x y -1 =y x -1 . The first condition implies that the graph has no loops. If an edge (x,y) exists between the two vertices x and y, then x≠y and x y -1 =y x -1 . But this also implies y x -1 = x y -1 and that the edge (y,x) also exists, with a unique edge incident (x,y) and (y,x) (we may assume the graph without multiple edges). This shows that the graph is without multiple edges and undirected.
2-To show the regularity of the graph, let x,y  X be two vertices of A 4 (G,X), then there is one to one correspondence between  1 (x) and  1 (y) that can be seen by looking to the map : 1 (x)  1 (y), which is defined as (a)=a g , for all a  1 (x) and g  G, which satisfies x g =y or g x g -1 =y (note that g always exists because x,y are conjugates in G). The map  is well-defined, since if a  1 (x) then a g y -1 = a g (x g ) -1 = g a x -1 g -1 = g x a -1 g -1 = y (a g ) -1 . This implies that a g   1 (y) . The map  is obviously one to one and onto. Note that the second property means that the choice of the fix t  X will be arbitrary. This is because of the regularity of the graph; each vertex has the same number of neighbors, so if we take s X and s≠t then | 1 (t)|= | 1 (s)|, and for any path in the graph that contains t, we can conjugate by g such that t g =s, then we obtain a path that contains s with the same length. We should also note that C G (t) is acting by conjugation on X. Also, if x is adjacent to y then for any w C G (x) we have x adjacent to y w . Thus, we can easily prove the following lemma: Lemma 1.4: ∆ i (t) of the A 4 (G,X) is a union of certain C G (t)-orbits.

Proof
Suppose that a  ∆ i (t) and w commute with t. We aim to show that a w  ∆ i (t). Since a  ∆ i (t), thus the path which contains a and t is of a length that is at most equal to i. Then, it is clear that if we conjugate this path by w we obtain a new path from a w and t is of a length that is at most equal to i. Thus, a w  ∆ i (t), as requested. For any group G and two subgroups H and K of G, the double coset of K in G is defined as the set HGK={h g k | h  H, g  G and k  K}. The number of C G (t)-orbits is called the permutation ranks of C G (t) on X. The next result shows the way of obtaining the size of C G (t)-orbits. Proposition 1.5: [8]. Suppose that G is a finite group and X is a conjugacy class of G. Then, the number of the C G (t)-orbits is equal to the number of (C G (t), C G (t))-double cosets. The above result does not only tell the permutation ranks. It also provides a representative for C G (t)orbits. For a G-conjugacy class C, define the set: X C ={x X| tx X}. One can see that if X C ≠Ø then it is equal to a union of certain C G (t)-orbits of X. The way of X C breaks into C G (t)-orbits. It will be essential to determine which discs of t contain the vertices in X C . Also, knowing the size of X C can be beneficial by leading to class structure constants. Class structure constants are the sizes of the sets: {(g 1 , g 2 )  C 1 x C 2 | g 1 g 2 = g} where C 1 , C 2 , C 3 are G-conjugacy classes and g is a fixed element of C 3 . Now, these constants can be calculated directly from the complex character table of G, which are recorded in the Atlas and are available electronically in the standard libraries of the computer algebra package Gap [9]. If we take C 1 = C, C 2 = X = C 3 and g = t, then in this case where h is a representative from C and 1 , 2 ,…, s are the complex irreducible characters of G. The next proposition gives the criteria to decide the order of tx if x  1 (t), which can be found by the below result: Proposition 1.6 1-Let x , y be any two distinct vertices in A 4 (G,X). Then if x is adjacent to y then the subgroup generated by x and y, <x,y>, isomorphic to A 4 , the alternating group of degree 4. 2-For x in  1 (t), we have that tx has the order 3. Proof 1-It is well known that the standard presentation of A 4 = <z,w|z 3 =w 3 , zwz=w -1 >. Now, x and y have the order 3 and x y -1 =y x -1 . Then, we have x y -1 x=y. If we set x=z and y -1 =w, we obtain that <x,y>A 4. 2-Since x  1 (t) then tx -1 =xt -1 , which leads to x=tx -1 t. As x,t have order 3, then we have (tx) 3 = tx tx tx= t tx -1 t t t x -1 t t tx -1 t = 1. This illustrates that tx has order 3.
These results of the next theorem were obtained computationally with the aid of Gap and the OnLine Atlas. In the context of these computations, we allocate the C G (t)-orbits on X. Representatives, in Gap format, for each of these orbits are to be obtained as downloadable files in [11], as they may be of value in other investigations of such group. In Section 3, we also give information on the action of C G (t) on X. Specially , we provide the C G (t)-orbit sizes for each X C ≠Ø. The main result of the paper is as follows. Theorem 2.2: Let G be isomorphic to 3 D 4 (2). Then 1-The sizes of the discs  i (t) are listed in Table 1 and the G-conjugacy classes of tx for x  i (t); i  ℕ are given in Table 2. 2-If (G,X)= ( 3 D 4 (2),3A)= ( 3 D 4 (2),3B), then Dim A 4 (G,X).= 5.

Proof
First, we fix an arbitrary element t in the class 3A or 3B and we set  0 (t)= t. Then, we calculate the C G (t)-orbits by using the Double Cosets of C G (t) in G, as we describe in section 1 and by Gap. Now, we break C G (t)-orbits into X C sets using the class representative from the OnLine Atlas. This can be seen in the below table.  (2). 3A  139776  27  648  13491  105463  20146  3B  326144  243  39852  285255  792  1 The above table proves that Dim A 4 (G,X)=5 for (G,X)=( 3 D 4 (2),3A)= ( 3 D 4 (2),3B).
For i≠0,  i (t) is equal to the set of each element in X C which adjacencies with some elements in  i-1 (t). We employ this property to obtain the following table below. As mentioned, we are using the class names in Atlas, although we have made some adjustments. First, we suppress the "slave" notation to write the class name of 3 D 4 (2). Second, for the purpose of simplification, we compress the letter part of the class name, since we aim to union these classes and their characters in alphabetical sequence. As in the example shown in Table 2, for G 3 D 4 (2) and X = 3A, 21AC is short-hand for 21A 21B21C .

C G (t)-Orbits On X
As mentioned above, we provide tables that include the sizes of the CG(t)-orbits, where C G (t) acts upon a non-empty set X C , with C is a G-conjugacy class. In the next tables, we employ an exponential notation to state the multiplicity of a certain size. For example, in the table for A 4 ( 3 D 4 (2),3A), the entry 216 2 ,378 next to 4A is implying that X 4A is the union of three CG(t)-orbits, two of which have the size of 216 and one has the size of 378. While, in the table for A 4 ( 3 D 4 (2),3B), the entry 324 4 ,648 14 next to 28AC indicates that each of X 28A , X 28B and X 28C is the union of eighteen C G (t)orbits, four of which have the size of 324 and fourteen have the size of 648. We give details of the permutation ranks in our next table. Table 3-Class sizes and Permutation Rank for A 4 ( 3 D 4 (2),X). In order to calculate the CG(t)-Orbits of A 4 ( 3 D 4 (2),3A) and A 4 ( 3 D 4 (2),3B), we present the following Procedure: Procedure 2 1. Choose t3A or 3B. 2. Compute Centralizer in G of t, C G (t).

A4-Graph |X=t G | Permutation Rank
3. Compute Double Cosets of C G (t) in G (C G (t)-orbits, which can be obtained from Proposition 1.5. 4. Break C G (t)-orbits into X C sets using the class representative from the OnLine Atlas. 5. Use the class structure constants to compute the size of X C.