Hurwitz groups and G_{2}(q)

Gunter Malle *, Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 288, D - 6900 Heidelberg.

Abstract: Finite factor groups of G_{2,3,7} := ⟨σ_{1}, σ_{2}, σ_{3} | σ_{1}^{2} = σ_{2}^{3} = σ_{3}^{7} = σ_{1}σ_{2}σ_{3} = ι⟩ are called Hurwitz groups. Here we prove that apart from ^{2}G_{2}(3), G_{2}(2), G_{2}(3) and G_{2}(4), all the groups ^{2}G_{2}(3^{2n+1}) and G_{2}(q), q = p^{n}, p ∈ ℙ, are Hurwitz groups. For the proof, (2, 3, 7) structure constants are calculated from the character tables [2], [7], and then with the lists of maximal subgroups [8], [5], the existence of generating triples is deduced.

Mathematics subject classification: 20F 05, 30F 35.

Hurwitz groups and G_{2}(q)

Gunter Malle *, Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 288, D - 6900 Heidelberg.

A (finite) group G is called a Hurwitz group if it is a factor group of

G_{2,3,7} := ⟨σ_{1}, σ_{2}, σ_{3} | σ_{1}^{2} = σ_{2}^{3} = σ_{3}^{7} = σ_{1}σ_{2}σ_{3} = ι⟩,

or, to put it differently, if it is generated by two elements of order two and three such that their product has order seven. The interest in such groups stems from the fact that they can be represented as a group of 84(γ − 1) automorphisms of a Riemann surface of genus γ. This is the biggest possible value for the order of a group acting on a surface of genus γ ≥ 2. As this bound was found by Hurwitz, such groups are called Hurwitz groups (see [9], and also [3] and the references cited there, for more recent investigations). The first example of such a group seems to have been found by Klein, namely G = L_{2}(7).

In [9], for example, the family of simple groups L_{2}(q) is studied and those of them which are Hurwitz groups are determined. Here we consider the Dickson-Cayley groups G_{2}(q), q = p^{n}, p ∈ ℙ. They should be good candidates for an investigation, since they all have orders divisible by 2, 3 and 7. In contrast to the situation in [9], where at most groups with q = p or q = p^{3} are Hurwitz groups, almost all of the G_{2}(q) turn out to be Hurwitz groups.

Theorem 1: Let q = p^{n}, p ∈ ℙ and q ≥ 5. Then G_{2}(q) is a Hurwitz group. On the other hand, none of G_{2}(2), G_{2}(3) or G_{2}(4) are Hurwitz groups.

Theorem 2: Let q = 3^{2m+1}, m ≥ 1. Then ^{2}G_{2}(q) is a Hurwitz group.

* The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft

* The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft

An immediate corollary of the two theorems is Corollary: All the groups G_{2}(q) and ^{2}G_{2}(q) are factor groups of the modular group SL_{2}(ℤ). Proof : The modular group SL_{2}(ℤ) is the free product of two cyclic groups of order two and three. Hence any group generated by two elements of orders two and three is a factor group. From the theorems, this is clear for G_{2}(q), q ≠ 2, 3, 4, and ^{2}G_{2}(q), q ≠ 3. (For the Ree groups, this already follows from Theorem 2.1 in [10].) With the help of the Atlas [4], it is not hard to check that (2B, 3B, 12C) of G_{2}(2), (2A, 3E, 13A) of G_{2}(3), (2B, 3B, 13A) of G_{2}(4) and (2A, 3B, 9D) of ^{2}G_{2}(3) contain generating triples for the respective groups. 1 Description of the method

The proof of the theorems is based on the knowledge of the character table and the maximal subgroups of G = G_{2}(q), which were calculated by several authors. To explain the strategy, we reformulate the condition for Hurwitz groups. Let G be a finite group, C_{1}, C_{2}, C_{3} three conjugacy classes, then call C = (C_{1}, C_{2}, C_{3}) a class structure, as in constructive Galois theory [11]. Also define

\overline{Σ}(C) = \{(σ_{1}, σ_{2}, σ_{3}) | σ_{i} ∈ C_{i}, σ_{1}σ_{2}σ_{3} = ι\}

and

Σ(C) = \{(σ_{1}, σ_{2}, σ_{3}) ∈ \overline{Σ}(C) | ⟨σ_{1}, σ_{2}, σ_{3}⟩ = G\} .

Then G is a Hurwitz group if there exists a class structure C = (C_{1}, C_{2}, C_{3}) such that C_{1} contains involutions, C_{2} and C_{3} elements of orders three, seven respectively, and |Σ(C)| > 0. We will consider triples in Σ(C) modulo conjugation in G. Then it is known that if Z(G) = 1 the normalized structure constant

n(C) := \frac{|G|}{|C_{G}(σ_{1})| · |C_{G}(σ_{2})| · |C_{G}(σ_{3})|} h ; ∑ ; i=1 \frac{χ_{i}(σ_{1})χ_{i}(σ_{2})χ_{i}(σ_{3})}{χ_{i}(ι)}

(where we sum over all irreducible characters χ_{i} of G) counts the number of classes of triples (σ_{1}, σ_{2}, σ_{3}) ∈ \overline{Σ}(C) modulo conjugation, each contributing (|Z(K)| · (N_{G}(K) : K))^{−1} to n(C) if K = ⟨σ_{1}, σ_{2}⟩ [11]. Hence if K_{1}, …, K_{s} is a full system of representatives of the conjugacy classes of maximal subgroups of G, we get that G is a Hurwitz group if

n_{G}(C) − s ; ∑ ; i=1 t_{i} ; ∑ ; j=1 n_{K_{i}}(C_{ij}) > 0,

where the second summation ranges over all class structures C_{ij} of K_{i} fusing into C. It is this condition which we will verify for G = G_{2}(q).

We now collect some information about suitable classes in the groups G_{2}(q). It can be deduced from the classification of the conjugacy classes in [1] and [6]. The abbreviation Φ_{i} = Φ_{i}(q) := i-th cyclotomic polynomial evaluated at q will be used throughout. Proposition 1: (a) Let p ≠ 7. Then G_{2}(q) contains a single class of regular elements of order seven.

(b) Let p ≠ 2. Then G_{2}(q) contains a single class of involutions.

(c) Let p ≠ 3. Then G_{2}(q) contains just two classes of elements of order three. All these classes are rational.

Proof : We have |G_{2}(q)| = q^{6}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}Φ_{6}. If q ≡ 2, 3, 4, 5 (mod 7) then Φ_{3}(q) or Φ_{6}(q) are divisible by seven, hence elements of order seven are contained in a maximal torus T of order Φ_{3}(q) or Φ_{6}(q). Apart from possibly elements of order three, all elements in those tori are regular. The Weyl group W (T) of those tori has order six, hence the elements of order seven are all conjugate. If q ≡ 1 (mod 7), then 7|Φ_{1}(q) and the maximal torus T ≅ Z_{q−1} × Z_{q−1} contains a Sylow 7-subgroup. From the action of the Weyl group W (G_{2}) ≅ D_{12} on T one sees that there exists exactly one class of regular elements of order seven. The same arguments apply for q ≡ −1 (mod 7) with T ≅ Z_{q+1} ×Z_{q+1}. Again from the action of the Weyl groups of the relevant tori, the assertion about the rationality of that class is deduced.

Parts (b) and (c) follow from the lists of conjugacy classes in [1] and [6]. 2 The general case

We now choose for p ≠ 7 the class 7A to contain the regular elements of order seven, for p ≠ 2 the class 2A to be the class of involutions, and for p ≠ 3 the class 3B to be the class of elements of order three with centralizer order qΦ_{1}Φ_{2}(q − ϵ), q ≡ ϵ (mod 3). If G_{2}(q_{0}) < G_{2}(q) with 𝔽_{q_{0}} < 𝔽_{q}, then it is clear from the description of the classes that exactly the three classes (2A, 3B, 7A) of G_{2}(q_{0}) fuse into (2A, 3B, 7A) of G_{2}(q). Proposition 2: Let (\underline{σ}) = (σ_{1}, σ_{2}, σ_{3}) be a (2, 3, 7)-system of G = G_{2}(q) with σ_{1}σ_{2}σ_{3} = ι. If 2 |̸ q, or 2|q and σ_{3} ∈ 7A, then ⟨\underline{σ}⟩ is not contained in a maximal parabolic subgroup of G. Proof : Maximal parabolic subgroups P of G_{2}(q) have the structure P = [q^{5}]: GL_{2}(q). If P has a (2, 3, 7)-system, then so has GL_{2}(q), the factor group of P by the unipotent radical. Inspection of the determinants shows that this triple even has to lie inside SL_{2}(q). But for 2 |̸ q this group does not have (2, 3, 7)-systems, since its only involution lies in the center. If 2|q, a different argument is needed. Here we use that no regular semisimple element (like σ_{3}) can lie inside the SL_{2}(q) of the factor group of P by the unipotent radical. This follows from the action of the Weyl group on the maximally split torus. Proposition 3: Theorem 1 holds for p ≠ 2, 3, 7 with C = (2A, 3B, 7A). Proof : Following the reformulated criterion, we first calculate the structure constant n_{G}(C). This is possible using the character table of G = G_{2}(q) in [2]. Different cases have to be distinguished, according to the congruences of q modulo 3 and 7. In Table 1, the respective values of the normalized structure constant n(2A, 3B, 7A) are displayed (the values where checked by Gerhard Hiss, using the MAPLE computer system):

Table 1: The (2, 3, 7) normalized structure constants in G_{2}(p^{n}), p ≠ 2, 3, 7.

q ≡  1 (mod 7)  −1 (mod 7) 2, 4 (mod 7) 3, 5 (mod 7)
1 (mod 3)  q^{2} + 3q + 1  q^{2} − q + 1  q^{2} + 1  q^{2} + 2q + 1
−1 (mod 3)  q^{2} + q + 1  q^{2} − 3q + 1  q^{2} − 2q + 1  q^{2} + 1

Taking into account the congruences for q this shows that for any q = p^{n}, p ≠ 2, 3, 7, we have \frac{1}{2}q^{2} ≤ n(C) ≤ 2q^{2}.

We must now decide which contribution to the structure constant comes from proper subgroups of G. By the considerations in the first part it is sufficient to do this for a representative system of classes of maximal subgroups. In Table 2 we list all classes of maximal subgroups K of G_{2}(q) for p ≥ 5 [8], except for the parabolic subgroups (they were excluded in Proposition 2).

Table 2: Some maximal subgroups of G_{2}(q), q = p^{n}, p ≥ 5.
K  occurs for  n_{K}(2, 3, 7)
(SL_{2}(q) ∘ SL_{2}(q)) · 2  always  ≤ 18
2^{3}^{.}L_{3}(2)  q = p  4
SL_{3}(q): 2  always  ≤ 3
SU_{3}(q): 2  always  ≤ 3
P GL_{2}(q)  p ≥ 7, q ≥ 11  ≤ 3
L_{2}(8)  𝔽_{q} = 𝔽_{p}[ζ_{7} + \overline{ζ_{7}}]  3
L_{2}(13)  𝔽_{q} = 𝔽_{p}[\sqrt{13}]  6
G_{2}(2)  q = p  1
J_{1}  q = 11  7
G_{2}(q_{0})  q = q_{0}^{t}, t ∈ ℙ  ≤ 2q_{0}^{2}

We have added a column showing the number of (2, 3, 7)-triples in K which can possibly fuse into our G-class structure C. For the involution centralizer we use that L_{2}(q) possesses at most six such classes, hence the central product can have no more than 36 (remember that the class 7A is regular). The 2 on top fuses them into at most 18. For SL_{3}(q) and SU_{3}(q) we could either invoke the lemma of Belyi [11], p.106, telling us that any class structure of SL_{3}(q) with an involution contains at most one generating class of triples, and then use the list of maximal subgroups of SL_{3}(q) to determine the contribution from proper subgroups of this maximal subgroup. But it is easier to explicitely calculate n_{SL_{3}}(C) from the character tables in [12].

The case of the involution centralizer contributes to n(C) only if seven divides its order, i.e. q ≡ ±1 (mod 7). The smallest such q in our analysis is q = 13. From the table and the value for n(C) above, the proposition immediately follows for q = 5. So we have q ≥ 11. Then the contribution from the maximal subgroups different from G_{2}(q_{0}) is at most 48. Hence for q = p, we are done. If q = p^{2}, the contribution from G_{2}(p) is at most 2p^{2}, and \frac{1}{2}p^{4} ≥ 2p^{2} + 48.

So q ≥ p^{3} ≥ 125. We can estimate 48 ≤ 2p^{2}. Let q = p^{n} and n = ∏_{i=1}^{r} t_{i}^{ν_{i}} the prime decomposition of n. Then we have to subtract at most

r ; ∑ ; i=1 2p^{2n/t_{i}} ≤ r ; ∑ ; i=1 2p^{n} = 2rp^{n}

as an upper bound for the part of n_{G}(C) lying inside all G_{2}(q_{0}). So the number l(C) of generating triples in the class structure (i.e. the order of Σ(C) modulo Inn(G)) is at least

l(C) ≥ \frac{1}{2}p^{2n} − 2rp^{n} − 2p^{2} ≥ \frac{1}{2}p^{n} (p^{n} − 4r − ϵ) ≥ 1, with an ϵ < 1,

for all prime powers under consideration. This proves the proposition. 3 The case p = 2

Let us now turn to the exceptional cases, namely p ∈ {2, 3, 7}. If p = 2, the involutions are unipotent. We take 2U to be the class of involutions of G_{2}(2^{n}) with centralizer order q^{4}Φ_{1}Φ_{2} [6].

Proposition 4: Theorem 1 holds for q = 2^{n} ≥ 8 with C = (2U, 3B, 7A).

Proof : The character values in G_{2}(2^{n}) may be calculated by the Deligne-Lusztig theory from the values of the Green functions. From them we obtain

Table 3: The (2, 3, 7) normalized structure constants in G_{2}(2^{n}).

q ≡  1 (mod 7) 2, 4 (mod 7)
1 (mod 3)  q^{2} + 3q  q^{2}
−1 (mod 3)  q^{2} + q  q^{2} − 2q

Hence we get \frac{1}{2}q^{2} ≤ n(C) ≤ 2q^{2} for all q = 2^{n}. As in the argument for p ≠ 2, 3, 7, we now need a list of maximal subgroups of G_{2}(2^{n}) [5] to see which part of the structure constant comes from proper subgroups.

Table 4: Some maximal subgroups of G_{2}(2^{n}), n ≥ 3.

K  occurs for  n_{K}(2, 3, 7)
L_{2}(q) × L_{2}(q)  always  ≤ 10
SL_{3}(q): 2  always  ≤ 3
SU_{3}(q): 2  always  ≤ 3
G_{2}(q_{0})  q = q_{0}^{t}, t ∈ ℙ  ≤ 2q_{0}^{2}

The maximal parabolics can be disregarded by Proposition 2. So, with the prime decomposition n = ∏_{i=1}^{r} t_{i}^{ν_{i}} we have

l(C) ≥ \frac{1}{2}q^{2} − 16 − r ; ∑ ; i=1 2 · 2^{\frac{2n}{ti}} ≥ 2^{2n−1} − 16 − r ; ∑ ; i=1 2^{n+1} = 2^{n+1}(2^{n−2} − r − 2^{3−n}) ≥ 1,

proving the assertion. (The case q = 8 needs a bit more careful estimations.)

4 The case p = 3

Next let p = 3. We will need the (2, 3, 7) structure constants of ^{2}G_{2}(3^{n}) to handle G_{2}(3^{n}). From them, it is immediate to verify

Proposition 5: Theorem 2 holds for ^{2}G_{2}(3^{2m+1}), m ≥ 1, with C = (2A, 3B, 7A).

Proof : The groups G = ^{2}G_{2}(q), q = 3^{2m+1}, possess one class 2A of involutions, one class 7A of (rational) elements of order seven and three classes of elements of order three [13]. We will choose 3B to be one of the two nonrational such classes. Then the character table of G in [13] yields \frac{1}{2}(q − \sqrt{3q}) ≤ n(C) ≤ \frac{1}{2}(q + \sqrt{3q}). The maximal subgroups of G are found in [8], and by similar but easier arguments to the G_{2}-cases, the proposition is deduced. Clearly, since (^{2}G_{2}(3))′ = L_{2}(8) < ^{2}G_{2}(3), this group can not be a Hurwitz group.

For G = G_{2}(3^{n}) it proves suitable to work with two classes of 3-elements. Namely, let 3U be the union of the two unipotent classes with representatives A_{41} and A_{42} in [7]. By abuse of notation, still write C = (2A, 3U, 7A), so that n(C) = n(2A, [A_{41}], 7A) + n(2A, [A_{42}], 7A). This is necessary because an element A_{42} of a subgroup G_{2}(3^{m}) < G_{2}(3^{n}) may fuse either into A_{41} or A_{42}, as is seen from the Steinberg representatives of the two classes in Proposition 6.4 in [6]. (There, the relevant elements are called x_{4} and x_{5}.)

Proposition 6: Theorem 1 holds for q = 3^{n} ≥ 9 with C = (2A, 3U, 7A).

Proof : From the character table of G_{2}(3^{n}) in [7] the following table of normalized structure constants is calculated:

Table 5: The (2, 3, 7) normalized structure constants in G_{2}(3^{n}).

q ≡ 1 (mod 7) −1 (mod 7) 2, 4 (mod 7) 3, 5 (mod 7) ; q^{2} + 2q  q^{2} − 2q  q^{2} − q  q^{2} + q

Hence once more we have \frac{1}{2}q^{2} ≤ n(C) ≤ 2q^{2} for q = 3^{n}. Next, the list of maximal subgroups, taken from [8], is as follows

Table 6: Some maximal subgroups of G_{2}(3^{n}), n ≥ 2.

K  occurs for  n_{K}(2, 3, 7)
(SL_{2}(q) ∘ SL_{2}(q)) · 2  always  ≤ 18
SL_{3}(q): 2  always  ≤ 1
SL_{3}(q): 2  always  ≤ 1
SU_{3}(q): 2  always  ≤ 1
SU_{3}(q): 2  always  ≤ 1
G_{2}(q_{0})  q = q_{0}^{t}, t ∈ ℙ  ≤ 2q_{0}^{2}
2G_{2}(q)  q = 3^{n}, n odd  ≤ 2q

The contribution from ^{2}G_{2}(q) is the sum over the three possible (2, 3, 7) structures of that group, see the proof of Proposition 5 and [13]. Estimations as in the preceeding sections now show that l(C) > 0. 5 The case p = 7

The regular unipotent elements of G_{2}(7^{n}) have order seven, as can be calculated from the Steinberg generators in [1], (2.1). (This contrasts to G_{2}(5^{n}), where those elements have order 25.) Call the class of regular unipotent elements 7U. Then by [1] or general theorems on regular unipotent elements in good characteristic, this is a rational class. Hence C = (2A, 3B, 7U) is a rational class structure,

Proposition 7: Theorem 1 holds for q = 7^{n} with C = (2A, 3B, 7U).

Proof : From the character table of G in [2] it is possible to calculate the following structure constants:

Table 7: The (2, 3, 7) normalized structure constants in G_{2}(7^{n}).

q ≡
1 (mod 3)  q^{2} + q
−1 (mod 3) q^{2} − q

The contribution from maximal subgroups of G was already collected in Table 2. (By Proposition 2, the maximal parabolic subgroups do not possess relevant (2, 3, 7)-systems.) Estimations as in the second section now show that there always remain generating triples for the whole group, and the proposition follows. 6 Exclusion of the remaining groups

Proposition 8: The groups G_{2}(2), G_{2}(3) and G_{2}(4) are no Hurwitz groups.

Proof : For G_{2}(2) = U_{3}(3): 2 this is trivial, since any group generated by a (2, 3, 7)-triple has to satisfy G = G′. It is less obvious for G_{2}(3) and G_{2}(4). Here we make heavy use of the Atlas tables for these groups and for various subgroups [4]. Moreover we use the fact that a class of (2, 3, 7)-systems generating a proper selfnormalizing subgroup H of G with trivial center contributes precisely 1 to the structure constant in G.

The group G_{2}(3) has five classes of elements of order three, 3A, …, 3E. The other relevant conjugacy classes were already mentioned in Proposition 1 (there exist no nonregular elements of order seven in G_{2}(3)). The nonzero structure constants are

n(2A, 3C, 7A) = 1, n(2A, 3D, 7A) = n(2A, 3E, 7A) = 6.

For the maximal subgroup L_{2}(8): 3, which contains elements from 2A, 3C and 7A, we find a (2, 3, 7)-structure constant of 1. This accounts for n(2A, 3C, 7A). Next L_{2}(13) has six classes of (2, 3, 7)-systems, and by [4] nontrivially intersects 3D, killing n(2A, 3D, 7A). For 2^{3}^{.}L_{3}(2), we know n(2, 3, 7) = 4, and as the extension is nonsplit (see also [8]), each triple has to generate all of 2^{3}^{.}L_{3}(2). Finally there are two classes of U_{3}(3): 2 in G_{2}(3), each having n(2, 3, 7) = 1. The group generated by these is at least an L_{2}(7). By comparing the fusion of elements of order four, they are seen to be distinct. So n(2A, 3E, 7A) also belongs to proper subgroups, and G_{2}(3) can not be a Hurwitz group.

The interesting conjugacy classes in G_{2}(4) are (in Atlas notation) two classes of involutions 2A, 2B, two classes of elements of order three 3A, 3B, and the class 7A. We get the structure constants

n(2A, 3A, 7A) = n(2B, 3A, 7A) = 0,
n(2A, 3B, 7A) = 1, n(2B, 3B, 7A) = 16.

The maximal subgroup J_{2} also has four interesting class structure with constants (the classes as in the Atlas)

n(2A, 3A, 7A) = n(2B, 3A, 7A) = 0,
n(2A, 3B, 7A) = 1, n(2B, 3B, 7A) = 10.

By inspection of the list of maximal subgroups of J_{2} and their fusion into J_{2} it is clear that all (2B, 3B, 7A)-triples generate J_{2}, giving a contribution of 10 to the structure constant of G_{2}(4). On the other hand, the (2A, 3B, 7A)-triple generates an L_{2}(7), which corresponds to the (2A, 3B, 7A)-triple in G_{2}(4). We next study the maximal subgroup L_{2}(13) of G. From Table 2 we see that this group has six (2, 3, 7)-systems, and they all generate the whole group. Hence another six of the (2B, 3B, 7A)-systems of G are taken away, and no one remains to generate all of G_{2}(4). This proves the assertion. Acknowledgement: I would like to thank Gerhard Hiss for checking the values of the G_{2}(q) structure constants, using the generic character tables in the MAPLE computer system.

References

[1] B. Chang, The conjugate classes of Chevalley groups of type (G_{2}), J. Algebra 9 (1968), p.190–211.

[2] B. Chang, R. Ree, The characters of G_{2}(q), Symposia Mathematica XIII, London 1974, p.395–413.

[3] M. Conder, A family of Hurwitz groups with non-trivial centres, Bull. Austr. Math. Soc. 33 (1986), p.123–130.

[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press 1985, Oxford.

[5] B. N. Cooperstein, Maximal subgroups of G_{2}(2^{n}), J. Algebra 70 (1981), p.23–36.

[6] H. Enomoto, The conjugacy classes of Chevalley groups of type (G_{2}) over finite fields of characteristic 2 or 3, J. Fac. Sci. Univ. Tokyo 16 (1970), p.497–512.

[7] H. Enomoto, The characters of the finite Chevalley groups G_{2}(q), q = 3^{f}, Japan J. Math. N.S. 2 (1976), p.191–248.

[8] P. Kleidman, The maximal subgroups of the Chevalley groups G_{2}(q) with q odd, the Ree groups ^{2}G_{2}(q) and of their automorphism groups, J. Algebra 117 (1988), p.30–71.

[9] A. M. Macbeath, Generators of the linear fractional groups, Proc. Symp. Pure Math. 12 (1969), p.14–32.

[10] G. Malle, Exceptional groups of Lie type as Galois groups, J. reine angew. Math. 392 (1988), p.70–109.

[11] B. H. Matzat, Konstruktive Galoistheorie, Springer Lecture Notes 1284, Berlin - Heidelberg - New York 1987.

[12] W. Simpson, J. Frame, The character tables for SL_{3}(q), SU_{3}(q), P SL_{3}(q), U_{3}(q), Can. J. Math. 25 (1973), p.486–494.

[13] H. N. Ward, On Ree’s series of simple groups, Trans. Am. Math. Soc. 121 (1966), p.62-89.