Commuting Involution Graphs for Certain Exceptional Groups of Lie Type

Suppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph C(G,X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}(G,X)$$\end{document} is the graph whose vertex set is X with x,y∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x, y \in X$$\end{document} being joined if x≠y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \ne y$$\end{document} and xy=yx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$xy = yx$$\end{document}. Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.


Introduction
Suppose that G is a finite group and X is a subset of G. The commuting graph, CðG; XÞ, has X as its vertex set and two vertices x; y 2 X are joined by an edge if x 6 ¼ y and x and y commute. The extensive bibliography in [9] points towards the many varied commuting graphs which have been studied. But here we shall be considering commuting involution graphs-these are commuting graphs CðG; XÞ where X is a G-conjugacy class of involutions. From now on X is assumed to be a Gconjugacy class of involutions. Because involutions are often centre stage in the study of non-abelian simple groups, there is a large literature on their commuting involution graphs. Indeed, such graphs have been instrumental in the construction of some of the sporadic simple groups. For example, the three Fischer groups with the conjugacy class being the 3-transpositions were investigated by Fischer [11], resulting in the construction of these groups. Later, also prior to their construction, commuting involution graphs for the Baby Monster (f3; 4g-transpositions) and the Monster (6-transpositions) were analyzed. Recently the commuting involution graphs of the sporadic simple groups have received much attention, see [5,12,14,15,17]. For those simple groups of Lie type consult [1,4,[8][9][10], while an analysis of the commuting involution graphs of finite Coxeter groups may be found in [2,3].
The aim of this short note is to describe certain features of CðG; XÞ when G is one of the exceptional Lie type groups of characteristic two. Specifically we consider G being one of the simple groups 3 D 4 ð2Þ; E 6 ð2Þ; 2 F 4 ð2Þ 0 and F 4 ð2Þ.
For x 2 X we define the i th disc of x, D i ðxÞ, (i 2 N) to be where dð; Þ is the usual distance metric on the graph CðG; XÞ. Of course, G acting by conjugation on X embeds G in the group of graph automorphisms of CðG; XÞ and, evidentily, G is transitive on the vertices of CðG; XÞ. We now choose t 2 X to be a fixed vertex of CðG; XÞ-our main focus is the description of the discs of t in CðG; XÞ. The diameter of CðG; XÞ will be denoted by Diam CðG; XÞand we shall rely upon the ATLAS [7] for the names of conjugacy classes of G. Our main result is as follows.

(i)
The sizes of the discs D i ðtÞ are listed in Table 1 and the G-conjugacy classes of tx for x 2 D i ðtÞ; i 2 N are given in Table 2  These results were obtained computationally with the aid of MAGMA [6] , GAP [16] and the ONLINE ATLAS [18]. In the course of these calculations we determined the C G ðtÞ-orbits on X (where C G ðtÞ is acting by conjugation). Representatives, in MAGMA format, for each of these orbits are to be found as downloadable files at [13], as they may be of value in other investigations of these groups. In Sect. 2 we also collate information on the action of C G ðtÞ on X. In particular, we give the C G ðtÞorbit sizes on each (non-empty) X C , X C being defined below.
We observe that some ''obvious'' groups are missing in this paper. First G 2 ð2Þ 0 being isomorphic to PSU 3 ð3Þ means it is covered in [8]. As for G ffi 2 E 6 ð2Þ, the cases X ¼ 2A and X ¼ 2B are done in [1], while there are partial results in the case X ¼ 2C. Likewise [1] also has partial results for E 7 ð2Þ. While E 8 ð2Þ is far and away beyond current computational capabilities.
We remark on the graphs studied here. First we note that as the outer automorphism of F 4 ð2Þ interchanges the two classes 2A and 2B, we have that CðF 4 ð2Þ; 2AÞ and CðF 4 ð2Þ; 2BÞ are isomorphic graphs. A very noteworthy consequence of the present work is that the distance between t and x in CðG; XÞ is almost always determined by the G-class to which tx belongs. The exceptions are G ffi Table 2 The conjugacy class of products tx for x 2 D i ðtÞ  Table 2 for more details-for example when G ffi F 4 ð2Þ; X ¼ 2D and tx 2 12I [ 12J each of 12I and 12J breaks into thirteen C G ðtÞ-orbits, 12 of size 294,912 and one of size 1,179,648 with those of size 294,912 being in D 2 ðtÞ and the one of size 1,179,648 in D 3 ðtÞ.
A word or two about the information in our tables is required. As mentioned we employ the class names given in the ATLAS though we make some modifications. First we suppress the ''slave'' notation. So, for example, the classes 7B Ã 2; 7C Ã 4 of 3 D 4 ð2Þ are just written as 7B, 7C, respectively. Secondly we compress the letter part of a class name when we mean the union of these classes and their letters are in alphabetical sequence. As an example, in Table 2 Let C be a G-conjugacy class and define It is clear that X C will either be empty or be a union of certain C G ðtÞ-orbits of X (where G acts upon X by conjugation). In locating which discs of t contain the vertices in X C we sometimes need to determine how X C breaks into C G ðtÞ-orbits.
Also of interest to us is the size of X C which leads us to class structure constants. Class structure constants are the sizes of sets 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 [16]. If we take C 1 ¼ C, C 2 ¼ X ¼ C 3 and g ¼ t, then in this case where h is a representative from C and v 1 ; . . .; v k the complex irreducible characters of G.
2 C G ðtÞ-Orbits on X As promised, we tabulate the sizes of the C G ðtÞ-orbits in their action upon X C where C is a G-conjugacy class for which X C is non-empty. In the ensuing tables we use an exponential notation to indicate the multiplicity of a particular size. Thus in the table for G ffi 3 D 4 ð2Þ with X ¼ 2B the entry 4 6 ; 24 12 next to 2B is telling us that X 2B is the union of eighteen C G ðtÞ-orbits, six of which have size 4 and twelve of which have size 24. Still looking at the same table, the entry 512, 1536 next to 9AC indicates that each of X 9A ; X 9B and X 9C is the union of two C G ðtÞ-orbits of sizes 512 and 1536. We give details of the permutation ranks in Table 3.