Commuting Involution Graphs for Certain Exceptional Groups of Lie Type

Abstract

Suppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph ${\mathcal {C}}(G,X)$ is the graph whose vertex set is X with $x, y \in X$ being joined if $x \ne y$ and $xy = yx$. Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.

1 Introduction

Suppose that G is a finite group and X is a subset of G. The commuting graph, ${\mathcal {C}}(G,X)$, has X as its vertex set and two vertices $x, y \in X$ are joined by an edge if $x \ne y$ 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 ${\mathcal {C}}(G,X)$ where X is a G-conjugacy class of involutions. From now on X is assumed to be a G-conjugacy 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 ($\{3,4\}$-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 ${\mathcal {C}}(G,X)$ when G is one of the exceptional Lie type groups of characteristic two. Specifically we consider G being one of the simple groups $^3D_4(2), E_6(2), ^2F_4(2)'$ and $F_4(2)$.

For $x\in X$ we define the $i^{\mathrm {th}}$ disc of x, $\Delta _i(x)$, ($i \in \mathbb {N}$) to be

$$\begin{aligned} \Delta _i(x) = \{y \in X \; | \; d(x,y)=i\} \end{aligned}$$

where $d( , )$ is the usual distance metric on the graph ${\mathcal {C}}(G,X)$. Of course, G acting by conjugation on X embeds G in the group of graph automorphisms of ${\mathcal {C}}(G,X)$ and, evidentily, G is transitive on the vertices of ${\mathcal {C}}(G,X)$. We now choose $t \in X$ to be a fixed vertex of ${\mathcal {C}}(G,X)$—our main focus is the description of the discs of t in ${\mathcal {C}}(G,X)$. The diameter of ${\mathcal {C}}(G,X)$ will be denoted by $\mathrm {Diam} \, {\mathcal {C}}(G,X)$and we shall rely upon the   Atlas [7] for the names of conjugacy classes of G. Our main result is as follows.

Theorem 1

Let G be isomorphic to one of $^3D_4(2), E_6(2), ^2F_4(2)'$ and $F_4(2)$.

(i):

The sizes of the discs $\Delta _i(t)$ are listed in Table 1and the G-conjugacy classes of tx for $x \in \Delta _i(t), i \in \mathbb {N}$ are given in Table 2.

(ii):

If $(G,X) = (E_6(2),2A), (E_6(2),2B), (^2F_4(2)',2A), (F_4(2),2A), (F_4(2),2B)$ or $(F_4(2),2C)$, then $\mathrm {Diam} \, {\mathcal {C}}(G,X)$= 2.

(iii):

If $(G,X) = (^3D_4(2),2A), (^3D_4(2),2B), (E_6(2),2C), (^2F_4(2)',2B)$ or $(F_4(2),2D)$, then $\mathrm {Diam} \, {\mathcal {C}}(G,X)$= 3.

Table 1 Disc sizes for ${\mathcal {C}}(G,X), G \cong \;^3D_4(2), E_6(2), ^2F_4(2)', F_4(2)$

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 $C_G(t)$-orbits on X (where $C_G(t)$ 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 $C_G(t)$ on X. In particular, we give the $C_G(t)$-orbit sizes on each (non-empty) $X_C$, $X_C$ being defined below.

We observe that some “obvious” groups are missing in this paper. First $G_2(2)'$ being isomorphic to $PSU_3(3)$ means it is covered in [8]. As for $G \cong \;^2E_6(2)$, the cases $X = 2A$ and $X=2B$ are done in [1], while there are partial results in the case $X=2C$. Likewise [1] also has partial results for $E_7(2)$. While $E_8(2)$ is far and away beyond current computational capabilities.

We remark on the graphs studied here. First we note that as the outer automorphism of $F_4(2)$ interchanges the two classes 2A and 2B, we have that ${\mathcal {C}}(F_4(2),2A)$ and ${\mathcal {C}}(F_4(2),2B)$ are isomorphic graphs. A very noteworthy consequence of the present work is that the distance between t and x in ${\mathcal {C}}(G,X)$ is almost always determined by the G-class to which tx belongs. The exceptions are $G\cong E_6(2), X=2C$ with $tx \in 12B \cup 12E \cup 12F$ and $G \cong F_4(2), X = 2D$ and $tx \in 12A \cup 12B \cup 12I \cup 12J.$ See Table 2 for more details—for example when $G \cong F_4(2), X=2D$ and $tx \in 12I \cup 12J$ each of 12I and 12J breaks into thirteen $C_G(t)$-orbits, 12 of size 294,912 and one of size 1,179,648 with those of size 294,912 being in $\Delta _2(t)$ and the one of size 1,179,648 in $\Delta _3(t)$.

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 $7B*2, 7C*4$ of $^3D_4(2)$ 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, for $G \cong F_4(2)$ and $X = 2D$, 8AF is short-hand for $8A \cup 8B \cup 8C \cup 8D \cup 8E \cup 8F$.

Let C be a G-conjugacy class and define

$$\begin{aligned} X_C = \{x\in X \; | \; tx \in C\}. \end{aligned}$$

It is clear that $X_C$ will either be empty or be a union of certain $C_G(t)$-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 $C_G(t)$-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

$$\begin{aligned} \{(g_1, g_2) \in C_1 \times C_2 \; | \; g_1g_2=g\} \end{aligned}$$

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 $C_1=C$, $C_2=X=C_3$ and $g=t$, then in this case

$$\begin{aligned} |X_C| = \frac{|G|}{|C_G(t)||C_G(h)|} \sum _{r=1}^k \frac{\chi _r(h)\chi _r(t)\overline{\chi _{r}(t)}}{\chi _r(1)}, \end{aligned}$$

where h is a representative from C and $\chi _1, \ldots , \chi _k$ the complex irreducible characters of G.

Table 2 The conjugacy class of products tx for $x \in \Delta _i(t)$

2 $C_G(t)$-Orbits on X

As promised, we tabulate the sizes of the $C_G(t)$-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 $G \cong \;^3D_4(2)$ with $X = 2B$ the entry $4^6,24^{12}$ next to 2B is telling us that $X_{2B}$ is the union of eighteen $C_G(t)$-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 $C_G(t)$-orbits of sizes 512 and 1536. We give details of the permutation ranks in Table 3.

Table 3 Class sizes and permutation rank

2.1 $G \cong \;^3D_4(2)$

$X = 2A$

2A

18

3A

512

4A

288

$X=2B$

2A	3, 24	2B	$4^6, 24^{12}$	3A	384	3B	512
4A	$24^5, 192$	4B	$24^{10}, 192$	4C	$384^6$	6A	1536
6B	$384^6$	7AC	512	7D	3072	8A	$384^6$
8B	$384^8$	9AC	512, 1536	12A	$1536^2$	13AC	3072
14AC	1536	18AC	$1536^2$	21AC	3072	28AC	$1536^2$

2.2 $G \cong E_6(2)$

$X = 2A$

2A

2790

2B

124992

3A

2097152

4B

2856960

$X=2B$

2A	63, $2160^2$	2B	$56, 4320, 30240^2,$ $30720^2, 64512,120960$	2C	$60480^2, 725760^2,$ 967680
3A	2359296	3B	16777216	4A	774144
4B	$725760, 967680^2,$ $2211840^2$	4C	$1935360^4, 3870720^4,$ $4423680^4, 7741440^2$	4D	$7864320^2, 8847360$
4E	$46448640^2$	4F	$2064384^2, 61931520^4$	4J	$123863040^2$
4K	743178240	5A	939524096	6A	$70778880^2$
6D	990904320	6F	1056964608	8C	$990904320^2$
12B	$1132462080^{2}$

$X = 2C$

2A	3, 84, 1536, 2016	2B	168, 224, 2016, 5376,  $8064^2,10752^2,16128,$ $32256^2,43008,86016$	2C	$96^2,5376,16128^3,32256^4$ $36864^4,64512^4,86016^4$ $129024^3,25048^3,1032192$
3A	917504, 1572864	3B	29360128	3C	134217728
4A	1536, 21504, 32256 $36864^3,64512^3,86016$ 786432, 1032192	4B	$1536^2,16128,32256^4$ $36864,43008,64512^4$ $86016,129024^3,258048^2$ $786432,1032192^3$	4C	$64512^4,129024^6,258048^2$ $516096^{12},688128^2,1032192^6$ $2064384^8,4128768^4$
4D	1032192, 1376256,  $2752512^2, 11010048,$ 16515072	4E	$258048^4,516096^{10},$ $1032192^{10},2064384^{12},$ $4128768^{22},8257536^2$ 33030144	4F	$1032192^{2},2064384^{2},$ $2752512^2, 4128768^{8},$ $5505024^2,16515072^8,$ $33030144^6$
4G	$4128768^{2}, 8257536^2,$ $16515072^4, 33030144^3,$ $66060288^2$	4H	$3748736^2,66060288^2$	4I	$11010048^2,16515072^2,$ $33030144^6, 66060288^7,$ 88080384, 264241152
4J	$1376256^2,2064384^{2}$, $4128768^{6}, 8257536^{16},$ $16515072^{10},33030144^{22}$ $66060288^{10}$	4K	$4128768,8257536^{6},$ $16515072^{13} , 33030144^{12},$ $66060288^{8}, 264241152$	5A	234881024, 1409286144
6A	$2752512,33030144^{3},$ $44040192^2$	6B	402653184	6C	528482304, 704643072
6D	$1835008, 66060288^{4},$ $88080384^3, 132120576^4,$ 176160768, 264241152,  528482304	6E	$37748736^2, 66060288^{2},$ $88080384^2,132120576^2,$ $528482304^2$	6F	88080384, 352321536 528482304, 704643072 1056964608
6G	2818572288	6H	$1056964608^2,4227858432$	6I	8455716864
7C	805306368	7D	3221225472	8A	$1572864^2, 33030144^{2},$ $37748736^2, 66060288^{2},$ $88080384^2, 132120576^2$ $528482304^2$
8B	$37748736^2, 44040192^2,$ $66060288^{2}, 88080384^2,$ $132120576^2, 528482304^2$	8C	$16515072^2, 33030144^{2},$ $66060288^{8} , 88080384^2,$ $132120576^{12}, 264241152^{10}$	8D	$132120576^2,264241152^{20}$ $1056964608^2$
8E	$176160768^2,528482304^4,$ $2113929216^2$	8F	$2113929216^5$	8G	$26441152^{4},528482304^4,$ $105664608^{12}$
8H	$2113929216^3$	8I	$427858432^6$	8J	$1056964608^{2} , 2113929216^6,$ $4227858432^4$
9A	22548578304	9B	3221225472, 9663676416	10A	$2818572288^2, 4227858432$
10B	$2818572288, 4227858432^2$ 8455716864, 16911433728	12A	$402653184^2$	12B	$132120576^2, 528482304^6$
12C	$264241152^8, 528482304^4$ $1056964608^{16}$	12D	$1409286144^2,2113929216^4,$ 4227858432	12E	$2818572288, 4227858432^3$
12F	4227858432, 8455716864	12G	5637144576	12H	$352321536^2, 2113929216^6$ $4227858432^4$
12I	$8455716864^2$	12J	$1056964608^{2}, 2113929216^6,$ $4227858432^8$	12K	$1409286144^2,4227858432^4$ $8455716864^4$
12L	16911433728	12M	$16911433728^2$	12P	$16911433728^2$
13A	19327352832	14G	16911433728	14H	9663676416
15C	22548578304	15D	7516192768, 22548578304	16A	$8455716864^4$
16C	$16911433728^4$	17A	45097156608	17B	45097156608
18A	$9663676416^2$	18B	67645734912	20A	$16911433728^2$
20B	$33822867456^4$	21G	19327352832	21H	45097156608
24A	$8455716864^8$	24B	$16911433728^4$	24C	$33822867456^2$
24D	$33822867456^2$	28K	$9663676416^2$	28L	33822867456
30E	$22548578304^2$	30F	67645734912

2.3 $G \cong \;^2F_4(2)'$

$X = 2A$

2A

10

2B

80

4C

640

5A

1024

$X = 2B$

2A	3, 12	2B	$12^3, 48^2$	3A	$256^2$	4A	$192^2$
4B	$96^2$	4C	$96, 192^2$	5A	768	6A	$768^2$
8CD	$384^2$	12AB	$768^2$	13AB	1536

2.4 $G \cong F_4(2)$

$X = 2A$

2A

270

2C

2016

3A

32768

4C

34560

$X = 2B$

2B

270

2C

2016

3A

32768

4D

34560

$X = 2C$

2AB	30	2C	$32^2,180, 1920^2$	2D	$720^2, 960^4, 11520$	3AB	32768
4AB	15360	4CD	11520	4F	$1024^2$	4JK	$30720^2$
4L	737280	4M	$184320^2$	5A	1048576	6GH	983040

$X = 2D$

2AB	$3, 12, 72^2, 192$	2C	$9, 12^2, 24^2, 72^4,$ $144^7, 192^2, 576^4$	2D	$24^4, 144^{29}, 576^{24}$ $1152^{16}, 9216$
3AB	2048, 6144, 24576	3C	262144	4AB	$192, 576^8, 1152^4$ 9216, 12288
4CD	$144^4, 192^3, 288^4$ $576^{13}, 1152^2, 2304^4,$ $4608^4,12288$	4EF	$576^4,1536^4$ $2304^4, 9216^8$	4GH	$2304^4, 4608^6, 9216^2$ $18432^4, 73728$
4I	$9216^{14}, 18432^8$ $36864^4, 73728^2$	4JK	$1152^4, 1536^4, 2304^4$ $4608^{20}, 9216^{16}, 18432^{22}$	4L	$9216^9, 36864^8$ $147456^2$
4M	$2304^{2}, 4608^{12}, 9216^{30}$ $18432^{36}, 36864^{2}$	4N	$147456^{4}$	4O	$36864^{12},147456^{4}$
5A	$196608^{2}, 589824$	6AB	$6144^{2}, 24576^2, 73728^3$	6CD	$36864^2, 49152^2$ $73728^3, 294912$
6EF	786432	6GH	$12288,36864^2,49152^2$ $73728^7,147456^2,294912$	6IJ	$73728^8,147456^4,294912^4$
6K	2359296	7AB	1572864	8A	$294912^4$
8B	$147456^8,294912^4$	8CF	$24576^2,73728^{10}$ $147456^4, 294912^4$	8G	$589824^2$
8HI	$294912^6$	8J	$589824^{16}$	8K	$589824^6$
9AB	1572864, 4718592	10AB	$589824^2, 1179648^2$	10C	$589824^2, 1179648^4$
12AB	$294912^4,1179648$	12CD	$786432^2$	12EH	$98304^2, 294912^4,589824^4$
12IJ	$294912^{12},1179648$	12KL	$2359296^2$	12MN	$589824^{14}$
12O	$2359296^4$	13A	9437184	14AB	4718592
15AB	1572864, 4718592	16AB	$2359296^4$	17AB	9437184
18AB	$4718592^2$	20AB	$2359296^4$	21AB	9437184
24AD	$2359296^4$	28AB	$4718592^2$	30AB	$4718592^2$