AbstractThis paper reports on a new and independent existence proof for the sporadic simple group Ly of Lyons, using only two permutations of degree 9 606 125, computed by Cooperman, Finkelstein, Tselman, and York. We will show that these two permutations generate a group G≃Ly, by first computing a base and strong generating set for G, and then checking the two hypotheses for Ly from Lyons’ original paper. Moreover, this produces a new presentation for Ly

Gollan, Holger W.

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doi:10.1006/jsco.2000.1010
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J. Symbolic Computation (2001) 31, 203–209
A New Existence Proof for Ly , the Sporadic Simple
Group of R. Lyons∗
HOLGER W. GOLLAN†
Computing Center, University of Essen, Germany
This paper reports on a new and independent existence proof for the sporadic sim-
ple group Ly of Lyons, using only two permutations of degree 9 606 125, computed by
Cooperman, Finkelstein, Tselman, and York. We will show that these two permutations
generate a group G ' Ly, by first computing a base and strong generating set for G,
and then checking the two hypotheses for Ly from Lyons’ original paper. Moreover, this
produces a new presentation for Ly.
c© 2001 Academic Press
1. Introduction
The classification of the finite simple groups is one of the outstanding mathematical
results in this century. One of the major parts deals with the sporadic groups, and
Gorenstein, Lyons and Solomon remark in their book (Gorenstein, 1982) that “...the
existence and uniqueness of the sporadic groups and the development of their properties
form a very elaborate chapter of simple group theory, spread across a large number of
articles. Moreover, some of the results are unpublished (e.g. Sims’ computer calculations
establishing the existence and uniqueness of the Lyons group Ly)”.
This paper reports on a new and independent existence proof for Lyons’ simple group
Ly, using only two permutations of degree 9606 125, computed by Cooperman et al.
(1997) and a lot of heavy computer calculations. These permutations have been com-
puted as a test case for an algorithm to compute permutation representations from
matrix representations (Cooperman et al., 1997), using the 111-dimensional matrix rep-
resentation of Ly over GF (5) that has been constructed in Meyer et al. (1985), and a
partially probabilistic algorithm to enumerate a conjugacy class of cyclic subgroups of
type 〈3A〉, knowing that there are 9606 125 of them, each with normalizer isomorphic to
3McL :2.
Our proof is done by checking the two hypotheses from Lyons’ original paper (Lyons,
1972) for the group G that is generated by these two permutations. Hence we have to
search for an involution t ∈ G such that
(i) C = CenG(t) ' 2A11 and
(ii) t /∈ Z∗(G).
∗This work was supported by the Deutsche Forschungsgemeinschaft under grants Go 487/3–1 and Go
487/3–2.
†E-mail: gollan@hrz.uni-essen.de
0747–7171/01/010203 + 07 $35.00/0 c© 2001 Academic Press
204 H. W. Gollan
To do this, we first go into the stabilizer of a point and construct a subgroup of type
3McL : 2. In the next step we prove that this subgroup is already the full stabilizer of a
point, therefore getting at least the correct order for our group G, by using a variation of
the well known Schreier–Sims method, making use of double cosets to reduce the number
of necessary tests. Finally, we check the two conditions above to prove that
G ' Ly.
To handle the heavy computer calculations, a standalone base and strong generating
set program, written by the author, was used. Moreover, we have used MAGMA (Bosma
et al., 1997) to do a lot of useful calculations in smaller degrees. All the computations
have been done on an IBM 9076–SP1 with eight nodes and some other IBM RS6000s
at the Institute for Experimental Mathematics. To keep this paper short, we will omit
nearly all the details of the calculations. They can be found either in Gollan (1995) or
Gollan (1998); for any questions the reader can consult the author via electronic mail.
Other existence proofs for Lyons’ simple group can be found in Gollan and Havas
(1999), using the original presentation of Sims (1972) and results from this paper, and
in the unpublished manuscript that was announced in Sims (1972).
2. The Double Coset Trick
We assume that the reader is familiar with the basic concepts of computational group
theory like orbits, Schreier vectors, bases, and strong generating sets. These go back to
the pioneering work of Sims (1970); a detailed description of the important algorithms
can, for example, be found in Butler’s lecture notes (Butler, 1991). One of the main tasks
in this area is the construction of a base and strong generating set for a given permutation
group to compute the order of the group in question and to set up the data for solving
a lot of interesting tasks. Usually, this is either solved by random methods, or with the
help of Schreier generators, a sufficient set of generators for the stabilizer of a base point.
Although the second method is deterministic, the number of Schreier generators equals
the number of generators for the full group times the index of the stabilizer (the length
of the orbit), which makes this method unusable for our purposes. Hence we will present
a modification of this method by using double cosets; this variation of the Schreier–Sims
method is a reinvention of old and unpublished ideas of Sims, that were presented by
him in several talks during the 1970s. Moreover, the Brownie–Cannon–Sims algorithm
is also based on these ideas. It was first implemented in 1986 and has been included in
CAYLEY (Cannon, 1984) and MAGMA (Bosma et al., 1997). Cannon and Havas (1992)
talk about this verification procedure that uses two point stabilizers, but again the details
are undocumented in the literature.
Algorithm 2.1. Let G be a group acting on the set Ω, generated by a set X. Fur-
thermore, fix a point β ∈ Ω, and let S = StabG(β) be the stabilizer of β in G. Given
a subgroup H ≤ S, the following will either prove that H = S, or it will produce an
element in S outside H.
(i) Let {γ1, . . . , γs} be a set of representatives for the different orbits of H on βG ⊆ Ω.
(ii) For 1 ≤ i ≤ s let ri be a fixed element of G with βri = γi. Then r1, . . . , rs are
representatives for different double cosets of H in G.
A New Existence Proof for Ly 205
(iii) For each γ ∈ βG there is a unique 1 ≤ i ≤ s such that γ ∈ γHi . Let gγ be a fixed
element of H with γ = γgγi . Note that β
rigγ = γ, hence different elements from βG
lead to different cosets of H.
(iv) Check whether
HriH =
⋃
γ∈γHi
Hrigγ
for all 1 ≤ i ≤ s. If this is true, then each of the double cosets is a union of cosets
with representatives from (iii), hence the cosets Hrigγ , 1 ≤ i ≤ s, γ ∈ γHi , form a
union of double cosets. Otherwise an element in S outside H is found.
(v) If (iv) is true, then for each of the cosets Hrigγ , 1 ≤ i ≤ s, γ ∈ γHi , and for each
of the generators t ∈ X, decide whether Hrigγt lies in one of the double cosets
Hr1H, . . . ,HrsH. If this is always true, then H = S; otherwise we find an element
in S that does not lie in H.
It is not hard to prove this algorithm, and at a first look it does not seem to do better
than the old approach. But the following remark will show that we get a big improvement
from using double cosets. To do this, we assume that the orbits are calculated in the usual
way with the help of Schreier vectors, i.e. each gγ is encoded in the Schreier vector as a
word in the generators of H, and deleting a generator from the right will again produce
another gδ.
Remark 2.2. (i) Part (v) of the algorithm becomes trivial, whenever the generator t is
an element of H already, since then each double coset is closed under multiplication
with t.
(ii) For any other generator t /∈ H, define
T = H ∩Ht−1 .
Then, for any y ∈ T , we have that yt ∈ H, hence there exists an h ∈ H with
yt = th.
As a consequence we see that, whenever a coset representative rigγ with γ ∈ γHi
ends with an element y ∈ T in his description in the Schreier vector, then it follows
that
rigγ = rigδy
with δ = γy
−1
and we have that
rigγt = rigδyt = rigδth,
hence rigγt lies in the same double coset as rigδt.
It follows that, besides the trivial coset, we only have to check those coset repre-
sentatives rigγ that do end with a generator outside T .
This remark will reduce the number of necessary tests dramatically; in our application
we will go from 9606 125 Schreier generators for each generator of the group to just 68
final relations that have to be checked.
Moreover, to prove the equality in part (iv) of the algorithm, it suffices to show that
H ∩Hri = StabH(γi), which can be further reduced to the proof of the statement that
StabH(γi) ≤ Hri , an easy membership problem.
206 H. W. Gollan
3. Inside the Stabilizer
In this section we will sketch the way to find a subgroup H ' 3McL : 2 inside the
stabilizer of a point in our group G. We will start with the two permutations t, z ∈
Sym9606 125, computed by Cooperman et al. (1997). While z has order 67 and no fixpoints,
the element t is an involution with 2685 fixpoints. We choose β1 to be the smallest fixpoint
of t and define
S = StabG(β1) ≤ G.
Computing the orbit of β1 under t and the powers of z, we easily find an element
x = ztz65tz34tz9 ∈ S
with order 10 and 5 fixpoints.
Defining
H = 〈t, x〉 ≤ S,
and computing the orbits of H on Ω, we get five of them, and exactly the suborbit sizes
of 3McL :2 in Ly as stated in the ATLAS (Conway et al., 1985), so we take this H as our
candidate. The rest of the section deals with the proof that this guess is correct, hence
H ' 3McL :2.
This proof is divided into three parts; first we will get to the sporadic group McL,
after that we will deal with the central part of order 3 and the automorphism of order 2.
In the next step we restrict our group H to one of the above orbits, namely the one of
length 15 400, since the central element of order 3 will fix this whole orbit. This eliminates
the central part and gives two permutations t˜ and x˜, and a group H˜ = 〈t˜, x˜〉 on 15 400
points with |H˜| = |McL : 2|. Looking at the character table of McL : 2 and the number
of fixpoints of t˜, which is 110, we see that t˜ cannot lie in McL, hence we have to search
for other elements. A few random calculations produce the elements
x˜1 = x˜2, x˜3 = (t˜x˜)3
with the property that
U˜ = 〈x˜1, x˜3〉
has order |U˜ | = |McL|.
It remains to show that U˜ is in fact isomorphic to McL. But it is easier to go directly
to the next step, lift x˜1 and x˜3 to x1 = x2, x3 = (tx)3, and try to prove that
U = 〈x1, x3〉 ' 3McL.
This can be done by finding elements inside U for the ATLAS presentation of 3McL; the
details of this computation can be found in Gollan (1998).
The only thing left in this section is to add our involution t to the group U and to show
that this gives a group of type 3McL : 2. This is done by showing that t normalizes U
with the help of a certain base and strong generating set for U . Moreover, these relations,
together with the relations from the ATLAS presentation for 3McL, give a presentation
for the group 3McL :2. Some additional computations finally prove that
H = 〈t, x〉 ' 3McL :2.
A New Existence Proof for Ly 207
4. The Order of the Group
The aim of this section is to compute the order of the group G by using the double
coset trick to prove that H ' 3McL :2 equals the stabilizer S.
To do this, we first compute a base of length 2 and a strong generating set for the
subgroup H, where the intermediate stabilizer T ≤ H has order |T | = 349 920. Then we
extend H to a subgroup
K = 〈H, y〉 ≤ G
by adding an additional involution y to H. Since H has five orbits on Ω, we get five orbit
representatives γ1, . . . , γ5 and five double cosets for Algorithm 2.1. For the four non-
trivial orbits we have to compute the stabilizers StabH(γi), where one of them equals T ,
and we have to use them to check part (iv) of the double coset trick. Again, we omit the
details here and refer the reader to Gollan (1998).
Looking at Remark 2.2 we note that y is the only generator of K outside our proposed
stabilizer H, hence we have to check only this generator and only coset representatives
that do end with a generator outside T = H∩Hy. But the number of such representatives
equals the number of orbits of T on Ω, and we can easily compute that we will end with
68 relations to check in part (v) of the double coset trick.
Producing these relations is an application of orbit calculations and Schreier vectors,
together with the use of the known base and strong generating set for H. As it turns
out, six of these relations are either trivial or direct consequences of the fact that y is
an involution. Once knowing these relations, it is a lengthy, but routine computation
with large permutations of degree 9606 125 to check that they are correct in our given
permutation group K. The interested reader can obtain the relations from the author;
they can also be found in Gollan (1998).
From this it follows that H ' 3McL :2 is the full stabilizer of β1 in K, hence
|K| = |Ly|.
In addition, it is not hard to prove that
K = G,
and therefore
|G| = |Ly|.
Collecting these 62 relations, together with the ones needed to check part (iv) of the
double coset trick and the relations for 3McL : 2 from Section 3, we get a presentation
for G, and, as we will prove in the next section, for Lyons’ simple group as well.
5. The End of the Proof
This final section finishes the existence proof for Lyons’ simple group by going back
to Lyons’ original paper (Lyons, 1972) and proving that the two assumptions needed
there are true in our group G. Thus we have to show that the centralizer of an involution
in G is isomorphic to the double cover of the alternating group A11, and that such an
involution does not lie in Z∗(G).
The first thing to notice is that, since H ' 3McL :2 has only two classes of involutions
and the index of H in G is odd, there can be at most two conjugacy classes of involutions
in G. Using a well known formula for induced characters, it follows easily that G has
208 H. W. Gollan
exactly one class of involutions, and we can take t as its representative. For the rest of
the paper let
C = CenG(t)
be the centralizer of t in G. Another application of the character formula gives the order
of C as
|C| = |2A11|,
and we are close to our final goal. To prove that
C ' 2A11,
we have to find elements in C for the classical presentation of 2A11 by Schur (1911).
Once again, details are in Gollan (1998). Knowing that the centralizer C has the right
isomorphism type, we are left with a proof for the fact that
t /∈ Z∗(G).
But this follows from Corollary 1 of Glauberman’s Z∗(G) paper (Glauberman, 1966),
since t is central in C and G has only one class of involutions. This proves that both
assumptions from Lyons’ original paper are true for our group G, and we have proved
the following theorem.
Theorem 5.1. G ' Ly, hence Lyons’ simple group exists.
Acknowledgements
The author thanks G. Michler for all his support, and G. Cooperman, W. Lempken,
and R. Staszewski for fruitful discussions and inspiring ideas.
Note added in proof: V. Gebhardt (Essen) has proved that the 62 non-trivial re-
lations from part (v) of the double coset trick can be replaced by just one additional
relation to produce a presentation for Ly in Section 4.
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Originally Received 15 August 1996
Accepted 20 April 2000


A New Existence Proof for Ly, the Sporadic Simple Group of R. Lyons 

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A New Existence Proof for Ly, the Sporadic Simple Group of R. Lyons

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