Algebra
Volume 2015 (2015), Article ID 587629, 15 pages
http://dx.doi.org/10.1155/2015/587629

Research Article

Involutions in the Automorphism Groups of Small Sporadic Simple Groups

Chris Bates, Peter Rowley, and Paul Taylor

School of Mathematics, The University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 6PL, UK

Received 31 July 2014; Accepted 26 November 2014

Academic Editor: Peter Fleischmann

Copyright © 2015 Chris Bates et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in standard generators.

1. Introduction

Groups permeate many areas of mathematics. Sometimes they have cameo roles; other times they are centre stage. Frequently it is involutions, elements of order two, that are in the spotlight. For instance, in the topological arena we have involutory maps on the -sphere in connection with the Smith Conjecture (see, e.g., [1–3]), while in Banach spaces we encounter such things as involutive gradings and fixed points of involutions (see [4, 5]). In areas of algebra, such as associative rings and algebraic groups, there are many sightings of involutions (see [6–8]). Involutions can often have a considerable influence on the structure of the group to which they belong. Even their absence can be telling—witness the Feit Thompson theorem [9]. For a finite group of even order, Brauer and Fowler [10] establish many results concerning involutions and other properties of the group. For example, they bound the index of a proper normal subgroup in terms of the number of involutions the group possesses. In a similar vein, for a finite group with at least two conjugacy classes of involutions the Thompson order formula ([11], Theorem 35.1) gives its order using data closely associated with the involutions. In the case when we have a finite nonabelian simple group, more often than not, its involutions play a dominant role (see, e.g., [12]).

This paper studies the involutions in where is a small sporadic finite simple group. By small we mean that is isomorphic to one of the following groups:

 , , , , , , , , , , , , , , .

The diminutive appellation aligns, more or less, with said group having a nontrivial permutation representation of degree at most 6156. Several of the larger sporadic groups have been studied individually in [13–16]. So for the remainder of this paper is assumed to be a small sporadic simple group and is a subgroup of containing . Also will denote an involution of . Put , the -conjugacy class of . Our aim is to study the suborbits of in its conjugation action on , or, in other words, to determine the action of on . This we do employing the services of the computational algebra packages Gap [17] and Magma [18] partnered by the electronic Atlas [19]. It goes without saying therefore that we use the Atlas notation and conventions as given in [20].

2. Calculating Orbit Representatives

As our starting point we take the smallest nontrivial permutation representation of as described in [19] with being generated by standard generators denoted here, as in [19], by and . In the case when , the standard generators for are, again as in [19], denoted by and . Having chosen a suitable element in to play the role of , we then forage for elements so that is a complete set of representatives for the -orbits of . In doing this we make frequent use of the standard command  IsConjugate(H,x,y)—this works effectively here since the degree of the permutation representation of is no more than 6156. Also observe that we may write our conjugating element as a word in and . Thus, our aim is to find which have relatively “small” length relative to the generating set for . However we may not always achieve the minimum possible length. In more detail we proceed as follows: define and for

For set

The main purpose of is to produce a colony of short words in and , a number of which may well yield the same element of . To speed matters up we prune out these duplicates. We hunt through for typically at most 20, so as to ensnare suitable conjugating elements . If this does not yield enough representatives for the -orbits, then we recalculate with and replaced by short words in and which are also generators for . For example, we might try replacing by and keeping the same, or try a more complicated substitution such as replacing with . Such a substitution was used to produce a in the case when and in the class , starting from the word . Some of the words for the may be further simplified and this was done by hand.

On a number of occasions this approach fails to deliver conjugating elements for some (often of small size) -orbits. To deal with such elusive -orbits, say , we begin by finding an in , usually by a random search. Then we obtain an element in such that . Now we run through some or all of the elements in , seeking a for which has small “length” (i.e., the total number of symbols in the given expression for ). Sometimes we may also vary . In somewhat more detail the Magma code employed is as shown in Algorithm 1.

F<a,b>:=FreeGroup(2);

phi:=hom< F->G  ∣  <a,G.1>,<b,G.2> >;

r,h:=IsConjugate(G,t,x);

sh:=h@@phi;

lengthmin:=#sh;

hmin:=h;

Ct:=Centralizer(G,t);

for c in Ct do

     sch:=(c*h)@@phi;

      if #sch lt lengthmin then

       lengthmin:=#(sch);

       hmin:=c*h; shmin:=sch;

     end if;

end for;

Algorithm 1

We remark that this procedure is not usually as efficient as using the set and moreover usually yields more complicated expressions for the conjugating elements.

We say a few words about how we guarantee that we have a representative of every orbit. By using the  IsConjugate(Ct,x,y) command, where is a fixed involution of the -conjugacy class and  Ct denotes , we ensure that no two representatives are in the same -orbit. We terminate our calculations for the -conjugacy class of when the sum of the -orbit sizes equals the size of the conjugacy class.

One may envision the information presented here to be useful in the following circumstances. Suppose is a finite group with . Further suppose that we have identified the subgroup as being isomorphic to via recognizing and as standard generators. (See [21] for a discussion of standard generators.) Then we may translate our information from to so that we see how acts upon its involution conjugacy classes within the group . Moreover, if is large from a computational standpoint (e.g., is large or is a matrix group of large degree), then having the as short words may be beneficial. To facilitate applications such as these, computer files containing the as words in and are available on request from the second author.

The following section gives the permutation ranks of on and the elements together with some additional information.

3. Orbit Representatives

So is a suitably chosen (and then fixed) involution of the conjugacy class of . For a -conjugacy class we define

Since is -invariant, will be a union of -orbits for each -conjugacy class for which . As is well known, may be calculated for any -conjugacy class using the complex character table (see, e.g., [22]), and this is easily carried out in Gap. As we proceed by breaking each (for ) into -orbits this is useful information.

Our first table gives an overview of the permutation ranks of in its action on . The succeeding tables consider in turn the possibilities for , with the first column identifying and the second the size of the -orbit (contained in ) and the third supplies a group element for which .

We emphasize that the following tables give the -orbits of —in the case when the tables are annotated so as to also yield the -orbits of . Before explaining how this is done, we remark that in all instances here when we have . Thus is a -conjugacy class and . Now suppose that and are two -conjugacy classes which fuse in (so is a -conjugacy class). Then for some and consequently . Hence, if is a -orbit contained in , will be a -orbit of . In this circumstance a broken horizontal line indicates that and are fused in and if is obtained by using the th listed conjugating element in then is obtained using the th listed conjugating element in . When a -conjugacy class is also a -conjugacy class, it may be the case that a -orbit contained in is the union of two -orbits in . A vertical line connecting two -orbits in signifies that their union is a -orbit.

3.1. Permutation Ranks

See Table 1.

Table 1

8

10
11
13

22

13
5
10

6
31
14

9

9
27
5
32

27
17

14
26

6
28

12
34
41

22
42

9
66
15
41

7
61

5
13
178

3.2.

3.2.1. ,

See Table 2.

Table 2

12

24

8

24

48

24

24

3.3.

3.3.1. ,

See Table 3.

Table 3

15

60

30

30

120

3.3.2. ,

See Table 4.

Table 4

96

3.3.3. ,

See Table 5.

Table 5

15

1

15

60

60

60

60

60

60

120

120

3.4.

3.4.1. ,

See Table 6.

Table 6

15

15

60

12

60

20

60

20

60

60

120

60

60

60

60

120

120

120

120

120

120

3.5.

3.5.1. ,

See Table 7.

Table 7

24

12

8

6

192

48

48

48

96

96

384

192

3.5.2. ,

See Table 8.

Table 8

42

7

112

168

3.5.3. ,

See Table 9.

Table 9

20

5

80

80

40

40

320

320

320

3.6.

3.6.1. ,

See Table 10.

Table 10

10

160

80

32

32

3.6.2. ,

See Table 11.

Table 11

15

24

60

60

60

60

120

24

120

24

120

240

60

60

60

60

120

120

120

120

240

240

3.6.3. ,

See Table 12.

Table 12

21

28

168

84

42

168

336

168

168

3.7.

3.7.1. ,

See Table 13.

Table 13

84

14

448

336

336

336

896

1344

3.8.

3.8.1. ,

See Table 14.

Table 14

80

30

640

480

960

128

1536

1920

3.8.2. ,

See Tables 15 and 16.

Table 15

45

30

360

24

180

360

720

Table 16

720

360

360

2880

720

720

720

720

2880

3.8.3. ,

See Table 17.

Table 17

105

28

336

630

3.8.4. ,

See Tables 18 and 19.

Table 18

60

5

60

240

160

120

240

120

30

120

120

192

192

320

1920

480

480

Table 19

480

480

1920

480

480

960

960

960

1920

1920

960

960

1920

960

960

960

960

3.9.

3.9.1. ,

See Table 20.

Table 20

120

10

192

640

480

80

960

320

960

320

960

960

960

1920

1920

1920

960

960

960

960

1920

1920

1920

1920

3.9.2. ,

See Table 21.

Table 21

153

272

612

306

1224

1224

816

816

816

2448

2448

2448

2448

3.10.

3.10.1. ,

See Tables 22 and 23.

Table 22

168

14

14

84

896

128

Table 23

112

112

1344

672

672

1792

5376

3.10.2. ,

See Table 24.

Table 24

60

15

120

80

2

320

960

120

120

480

240

240

960

960

160

160

1920

1920

1920

1920

1920

1920

7680

3840

3840

3.11.

3.11.1. ,

See Table 25.

Table 25

210

2240

5040

8064

6720

3.11.2. ,

See Table 26.

Table 26

165

220

660

220

1980

990

792

792

7920

1980

1980

1980

3960

3960

3960

3960

3960

3960

2640

2640

3960

3960

3960

3960

3960

3960

3960

3960

3960

3960

3960

3960

3960

3960

3.12.

3.12.1. ,

See Table 27.

Table 27

240

2

315

1344

960

480

480

2688

13440

3.12.2. ,

See Table 28.

Table 28

84

896

2688

5376

5376

3072

21504

21504

21504

21504

21504

3.12.3. ,

See Table 29.

Table 29

105

63

315

630

126

2520

1260

1260

2520

3780

1890

1890

1260

7560

1890

90

1890

90

2160

7560

1080

7560

1080

7560

7560

7560

3780

3780

1890

1890

3780

3780

1890

1890

7560

7560

7560

7560

15120

15120

15120

15120

15120

7560

7560

7560

7560

3.13.

3.13.1. ,

See Table 30.

Table 30

640

480

240

30

10240

5120

7680

3840

3840

15360

7680

12288

4096

61440

30720

61440

61440

61440

61440

122880

61440

3.13.2. ,

See Table 31.

Table 31

455

455

455

1

1

5824

3640

3640

3640

5824

5824

29120

29120

29120

29120

29120

29120

29120

29120

29120

58240

29120

29120

29120

29120

29120

29120

29120

29120

29120

116480

58240

58240

29120

29120

29120

29120

29120

29120

116480

116480

3.14.

3.14.1. ,

See Table 32.

Table 32

360

54

5120

9216

1728

17280

55296

46080

3.14.2. ,

See Table 33.

Table 33

315

630

560

2

224

2240

20160

2520

1260

2520

2520

1260

1260

5040

2520

2520

10080

10080

1120

1120

20160

40320

10080

20160

40320

40320

80640

20160

20160

20160

20160

20160

20160

10080

10080

20160

20160

20160

20160

20160

20160

40320

40320

161280

40320

20160

40320

80640

80640

161280

161280

80640

161280

161280

161280

161280

161280

80640

80640

3.14.3. ,

See Table 34.

Table 34

315

1800

200

16800

3150

630

12600

20160

12600

100800

100800

25200

25200

50400

3.14.4. ,

See Table 35.

Table 35

495

792

264

5280

1760

2640

2970

11880

11880

19008

47520

11880

15840

15840

15840

15840

47520

15840

15840

47520

95040

31680

95040

31680

95040

47520

47520

190080

47520

47520

47520

47520

47520

190080

95040

95040

95040

95040

3.15.

3.15.1. ,

See Tables 36 and 37.

Table 36

630

8960

1920

Table 37

30240

48384

80640

3.15.2. ,

See Tables 38 and 39.

Table 38

495

396

396

1320

440

2640

1584

7920

2970

11880

5940

2970

9504

9504

47520

11880

3960

15840

15840

23760

23760

15840

23760

23760

95040

31680

23760

23760

23760

23760

Table 39

23760

23760

23760

23760

31680

31680

63360

47520

47520

47520

47520

23760

23760

47520

47520

47520

95040

95040

95040

95040

190080

95040

95040

190080

95040

95040

95040

95040

95040

95040

3.16.

3.16.1. ,

See Table 40.

Table 40

1260

1008

14336

40320

3.16.2. ,

See Table 41.

Table 41

15

1680

210

2520

35840

1920

20160

161280

13440

13440

344064

430080

3.16.3. ,

See Table 42.

Table 42

240

45

720

240

180

30

1440

1440

1440

1152

360

10240

15360

7680

1024

3840

1440

11520

5760

5760

1440

1440

720

720

11520

11520

5760

5760

2880

1440

1440

720

720

23040

11520

5760

2880

2880

23040

23040

11520

11520

5760

5760

5760

23040

23040

11520

11520

5760

5760

5760

5760

5760

5760

2880

2880

1920

1920

46080

46080

23040

23040

73728

184320

122880

92160

122880

46080

46080

15360

15360

92160

23040

15360

15360

92160

92160

46080

46080

46080

46080

15360

92160

92160

46080

46080

46080

46080

737280

92160

92160

92160

92160

46080

46080

23040

23040

92160

92160

92160

92160

92160

92160

92160

92160

92160

92160

92160

46080

46080

46080

46080

46080

46080

46080

46080

23040

23040

184320

184320

92160

92160

737280

245760

368640

368640

368640

368640

368640

1474560

122880

122880

368640

184320

368640

184320

184320

184320

92160

92160

368640

368640

184320

184320

184320

184320

92160

92160

368640

184320

184320

184320

184320

184320

184320

737280

737280

737280

368640

368640

368640

368640

737280

737280

737280

737280

737280

737280

737280

737280

737280

737280

737280

737280

737280

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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