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 
3.2.
3.2.1. ,

See Table 2.

Table 2 

3.3.
3.3.1. ,

See Table 3.

Table 3 
3.3.2. ,

See Table 4.

Table 4 
3.3.3. ,

See Table 5.

Table 5 
3.4.
3.4.1. ,

See Table 6.

Table 6 
3.5.
3.5.1. ,

See Table 7.

Table 7 
3.5.2. ,

See Table 8.

Table 8 
3.5.3. ,

See Table 9.

Table 9 
3.6.
3.6.1. ,

See Table 10.

Table 10 
3.6.2. ,

See Table 11.

Table 11 
3.6.3. ,

See Table 12.

Table 12 
3.7.
3.7.1. ,

See Table 13.

Table 13 
3.8.
3.8.1. ,

See Table 14.

Table 14 

3.8.2. ,

See Tables 15 and 16.

Table 15 
Table 16 
3.8.3. ,

See Table 17.

Table 17 
3.8.4. ,

See Tables 18 and 19.

Table 18 
Table 19 
3.9.
3.9.1. ,

See Table 20.

Table 20 
3.9.2. ,

See Table 21.

Table 21 
3.10.
3.10.1. ,

See Tables 22 and 23.

Table 22 
Table 23 
3.10.2. ,

See Table 24.

Table 24 
3.11.
3.11.1. ,

See Table 25.

Table 25 
3.11.2. ,

See Table 26.

Table 26 

3.12.
3.12.1. ,

See Table 27.

Table 27 
3.12.2. ,

See Table 28.

Table 28 

3.12.3. ,

See Table 29.

Table 29 
3.13.
3.13.1. ,

See Table 30.

Table 30 
3.13.2. ,

See Table 31.

Table 31 

3.14.
3.14.1. ,

See Table 32.

Table 32 
3.14.2. ,

See Table 33.

Table 33 
3.14.3. ,

See Table 34.

Table 34 
3.14.4. ,

See Table 35.

Table 35 
3.15.
3.15.1. ,

See Tables 36 and 37.

Table 36 
Table 37 

3.15.2. ,

See Tables 38 and 39.

Table 38 
Table 39 
3.16.
3.16.1. ,

See Table 40.

Table 40 
3.16.2. ,

See Table 41.

Table 41 
3.16.3. ,

See Table 42.

Table 42 

Conflict of Interests

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