A note on involution centralizers in black box groups

2.1 Involutions in the Lyons group

The sporadic simple group Ly, first pin-pointed by Lyons in [15], has order 28⋅37⋅56⋅7⋅11⋅31⋅37⋅67. Its smallest non-trivial permutation degree is 8,835,156 – indeed, this representation was built in the 1970s by Sims [22] in the course of showing the existence of Ly. Despite considerable advances in computer algebra, such a permutation representation is not viable for probing the structure of Ly. More tractable is the 111-dimensional representation over GF⁢(5), first displayed in [17], which we employ here. From [24], we obtain this representation where Ly is generated by the standard generators a,b with a∈2⁢A, b∈5⁢A and a⁢b∈14⁢A (using Atlas [10] names for the conjugacy classes of Ly).

For many of the sporadic simple groups G, the action (by conjugation) of G on a G-conjugacy class of involutions has been analyzed. See [4, 5, 21, 23], where the suborbits are determined. This type of information is valuable when studying such topics as commuting involution graphs [3], C-strings and polytopes [16].

Here we investigate the action of G≅Ly upon X=2⁢A. Before stating our main conclusion, we introduce two pieces of notation. Let t∈X. For C a G-conjugacy class, XC={x∈X∣t⁢x∈C}, and for y∈G,

Observe that XC will either be empty or the union of CG⁢(t)-orbits (under conjugation) and, of course, X=⋃˙XC, as C runs through all the G-conjugacy classes. As is well known, |XC| equals a particular structure constant which may be calculated using the complex irreducible character table of G. Also, we have [CG*(y):CG(y)]=1 or 2.

The following easy lemma will be used in the proof of Theorem 2.2.

So, for z=t⁢x∈C, where x∈X if we know the structure of CG⁢(z), we may be able to calculate CG⁢(t)∩CG⁢(x). In particular, note that [CG*(z):CG(z)]=2 and, as G=Ly has just one conjugacy class of involutions, CCG⁢(z)⁢(y) as y ranges through all involutions in CG*⁢(z)\CG⁢(z) gives all the possible structures for

that must occur (but not the possible multiplicities).

As we shall see, a number of the XC may be easily broken into CG⁢(t)-orbits. However, for three G-conjugacy classes, it is not so straightforward, and we make use of the routine (1), (2′), (3′), (4′), (5′) given in Section 1.

2.2 Calculations in E6⁢(2)

We now look further at the calculation in E6⁢(2), mentioned in Section 1. Repeating the calculation for the given t and s 100 times, in each case using 10,000 random elements, gave the following five outcomes with the following frequencies.

On the other hand, using 12,000 random elements from E6⁢(2) in each of 100 experiments always gave |〈S〉|=212⋅3⋅7.

2.3 Calculations in E8⁢(2)

We again go to [24] for our copy of G=E8⁢(2), using the 2-generator version there. So G=〈a,b〉 with a and b being 248×248 matrices over GF⁢(2). Let V denote the 248-dimensional GF⁢(2)⁢G-module. The order of a is 30 and the order of b is 10. We shall employ the naming system in [1] (which follows Atlas conventions) for the conjugacy classes of G. So a15∈2⁢D and b5∈2⁢B (because dim⁡CV⁢(a15)=128 and dim⁡CV⁢(b5)=156). In each of the following cases, we did 10 experiments each using 40,000 random elements from E8⁢(2). (i) t=a15 and s=tg, where g=b3⁢a⁢b-1⁢a⁢b⁢a11. Then ts has order 17 with dim⁡CV⁢(t⁢s)=32, so, using [1], t⁢s∈17⁢A⁢B with CG⁢(t⁢s)≅17×Ω8-⁢(2).

(ii) t=b5 and s=tg, where g=a⁢b3⁢a7⁢b4⁢a19. This time ts has order 8.

In case we are getting too rosy a picture, we have the following.

1. t=b5 and s=tg, g=a⁢b3⁢a7. Here ts has order 5, dim⁡CV⁢(t⁢s)=68. So t⁢s∈5⁢A with CG⁢(t⁢s)≅5×Ω12-⁢(2). Each of the 10 experiments yielded no elements of S at all. Moreover, note that, by Lemma 2.1, CG⁢(t)∩CG⁢(s) must be non-trivial.

2.4 Centralizers in a 670×670 matrix group

Using the GF⁢(2) representation of O10+⁢(2) of degree 670 given in [24], we may regard G=O10+⁢(2) as a subgroup of GL670⁢(2). In the notation of [24], G=〈a,b〉, where a has order 2 and b has order 20. Set H=〈a,ab,a(b⁢a⁢b3)〉. We have that H≅L3⁢(2), and we seek to calculate CG⁢(H). First employing the procedure in Section 1, we calculate CG⁢(〈a,ab〉) (which has order 27⋅33⋅5⋅7). Then using random elements selected from CG⁢(〈a,ab〉), we repeat the procedure to calculate elements in CCG⁢(〈a,ab〉)⁢(a(b⁢a⁢b3)) (and note that a(b⁢a⁢b3)∉CG⁢(〈a,ab〉)), thus delivering a subgroup of CG⁢(H) of order 22⋅32. There is of course the perennial problem of knowing whether we have obtained all of CG⁢(H) – though there are instances where just having some elements of CG⁢(H) may be helpful. As it happens, |CG⁢(H)|=22⋅32 (which may be checked by working in the 496 degree permutation representation for O10+⁢(2) available in [24]). Finally, we remark that the standard Magma functions could not calculate CG⁢(H).