Generation of classical groups

Gunter Malle *, Mathematisches Institut und IWR, Im Neuenheimer Feld 368, 69 Heidelberg, Germany

Jan Saxl ^{∗}, DPMMS, 16 Mill Lane, Cambridge CB2 1SB, England

Thomas Weigel ^{∗}, Mathematisches Institut, Albertstraße 23b, 78 Freiburg, Germany

The search for special small generating sets for finite simple groups has been a fruitful area of research for a number of years now, with a broad spectrum of applications. It is well known that every finite simple group is generated by two of its elements. In constructive Galois theory, generation by a rigid triple of its elements is a sufficient condition for Galois realizations with nice properties. The homomorphic images of P SL_{2}(ℤ) are generated by a 2-element and a 3-element. In the case of Hurwitz groups, one is searching for two generators of orders 2 and 3 with product of order 7.

A natural question in the area is to find the minimum number i(G) of involutions generating the simple group G. Of course i(G) is at least three, since any two involutions generate a dihedral group. In this paper we conclude the proof that, with one exception, three involutions always suffice:

Theorem A: Every finite simple non-abelian group other than U_{3}(3) is generated by three involutions.

The fact that i(U_{3}(3)) = 4 was proved by Wagner in [34], so this is a genuine exception. The Theorem had been proved previously in a large number of cases: alternating and linear groups, symplectic groups of dimension at least 6 and the Suzuki groups in [17], S_{4} and U_{3} in [10, 34], orthogonal groups in [11, 8, 9], classical groups of dimension at least 5 in even characteristic in [32], sporadic groups in [7], and finally the third author [36] dealt with exceptional groups of Lie type other than the two families of type E_{6}. In this paper we conclude the proof by presenting a unified approach for all the classical groups and by dealing with the groups of types E_{6}.

Our proof, which is based on the ideas in [36], has two main ingredients. Firstly, we use the classification of finite simple groups to obtain information on subgroups of classical groups containing a certain semisimple element of large order; this part expands the work of Dempwolff [12]. Secondly, we use the representation theory of groups of Lie type to obtain information concerning products of certain two conjugacy classes of elements of our group G. Putting these together (and using some of the above mentioned results of other authors), we obtain the following result (which clearly implies Theorem A).

Theorem B: Every finite non-abelian simple group other than U_{3}(3) is generated by two elements, one of which is strongly real and the other is an involution.

To conclude the introduction, we mention two related open questions: find the minimum number i_{c}(G) of conjugate involutions generating the non-abelian simple group G; and, find the number i_{c,1}(G) of conjugate

* The authors thank the MSRI in Berkeley, where this work was begun in fall 1990, for its hospitality. The first and third author gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft involutions generating G subject to the further condition that their product is 1. We remark that i_{c}(G) = 3 for a number of families of G, but i_{c}(A_{8}) = 4. We also remark that i_{c,1} ≥ 5 and i_{c,1}(A_{7}) = 6.

1 Maximal subgroups containing a certain semisimple element

In the following let G̃ be a finite perfect classical group which acts faithfully on its natural module V. Thus if G̃ is of type A_{l}, ^{2}A_{l} or C_{l}, G̃ = (G_{sc})^{F} is the fixed point set of some Frobenius morphism F: G_{sc} ↦ G_{sc} of a simply connected simple algebraic group G_{sc}. If G̃ is of type B_{l}, D_{l} or ^{2}D_{l}, G̃ is the homomorphic image of ϕ: (G_{sc})^{F} ↦ G̃, where ker(ϕ) has order 2 or 1. So the groups we are considering are the following: G̃ = SL_{k}(q), SU_{k}(q), Sp_{2k}(q), Ω_{k}^{±}(q).

Further we consider a semisimple element s̃ ∈ G̃ of a certain maximal torus T̃, where the type of T̃ is listed in Table 1.1 (for the notation we refer to [4]). So the type of T̃ corresponds to the cycle type of the W.F -conjugacy class of twisting elements of the (G_{sc})^{F} -conjugacy class of T̃. The element s̃ ∈ T̃ is chosen such that S̃ := ⟨s̃⟩ has the following properties: Let K = Hom_{𝔽_{p}G̃}(V, V). Then V ≃_{KS̃} W ⊕ W_{1}, where W = [V, S̃] is an irreducible KS̃-module and W_{1} = C_{V} (S̃) is a trivial KS̃-module. Thus S̃ is a group which is called a 1-irreducible subgroup of GL_{K}(V) [12]. In Table 1.1 we have listed the order of S̃ and the splitting of V as KS̃-module (last column).

Let ψ: G̃ ↦G̃ /Z(G̃) be the natural projection on the simple group G̃ /Z(G̃). We define G = ψ(G̃), S = ψ(S̃), s = ψ(s̃), …etc.

Table 1.1

G  type of T̃  |T̃|  |S̃|  dim_{K}(V)
A_{l} l even  (l + 1)  \frac{q^{l+1}−1}{q−1}  |T̃|  l + 1
l ≥ 3, l odd  (l)  q^{l} − 1  \frac{q^{l}−1}{q−1}  l + 1 = l ⊕ 1
2A_{l} l even, (l, q) ≠ (2, 2)  (l + 1)  \frac{q^{l+1}+1}{q+1}  |T̃|  l + 1
l ≥ 3, l odd  (l)  q^{l} + 1  \frac{q^{l}+1}{q+1}  l + 1 = l ⊕ 1
B_{l} l ≥ 3, q odd  (−, (l))  ^{1}/_{2}(q^{l} + 1)  |T̃|  2l + 1 = 2l ⊕ 1
C_{l} l ≥ 2, (l, q) ≠ (2, 2)  (−, (l))  q^{l} + 1  |T̃|  2l
D_{l} l ≥ 4  (−, (l − 1)(1)) ^{1}/_{d}(q^{l−1} + 1)(q + 1)^{∗} ^{1}/_{d}(q^{l−1} + 1) 2l = 2(l − 1) ⊕ 2
2D_{l} l ≥ 4  (−, (l))  ^{1}/_{d}(q^{l} + 1)^{∗}  |T̃|  2l
∗ d = gcd(2, q − 1)

If G is of type A_{l} or ^{2}A_{l} with l even, C_{l} or ^{2}D_{l}, T̃ is a cyclic torus and V is an irreducible KT̃-module. If G is of type A_{l} or ^{2}A_{l} with l odd, B_{l} or D_{l}, we use the natural embeddings A_{l−1}(q) < A_{l}(q), ^{2}A_{l−1}(q) < ^{2}A_{l}(q), 2D_{l−1}(q) < B_{l}(q) and ^{2}D_{l−1}(q) ≤ D_{l}(q) to show the existence of an element s̃ ∈ T̃ with the described properties.

Theorem 1.1: Let M be a maximal subgroup of the finite simple classical group of Lie type G containing the semisimple element s ∈ T of a maximal torus T of G of type listed in Table 1.1. Then one of the following holds:

(a) M ∈ C_{1} ∪ … ∪ C_{8} as defined in [1]; more precisely we have:

type(G) A_{l} (l even) A_{l} (l odd) ^{2}A_{l} (l even) ^{2}A_{l} (l odd) B_{l}  C_{l}  D_{l}  ^{2}D_{l}
M ∈  C_{3}  C_{1}  C_{3}  C_{1}  C_{1} C_{3} ∪ C_{8} C_{1} ∪ C_{3} C_{3}

(b) M ∈ S as defined in [1] and either

(i) (M, G) ≃ (A_{7}, A_{3}(2)), (A_{1}(7),^{2}A_{2}(3)), (A_{7},^{2}A_{2}(5)), (A_{7},^{2}A_{3}(3)), (A_{2}(4),^{2}A_{3}(3)), (A_{1}(11),^{2}A_{4}(2)),

(M_{22},^{2}A_{5}(2)), (S_{9}, B_{3}(3)), (A_{1}(17), C_{4}(2)), or (ii) (F^{∗}(M), G) = (B_{3}(q), D_{4}(q)) with q odd, or (C_{3}(q), D_{4}(q)) with q even.

(c) G ≃ A_{1}(q),^{2}A_{3}(2) or one of the following holds: (M, G) ≃ (A_{2}(2), A_{2}(4)), (2^{4}.A_{5}, C_{2}(3)), (A_{9}, D_{4}(2)).

Proof : We may assume that G is not of type A_{1}. To avoid the problems arising from the exception of the assertion of Zsigmondy’s Theorem we first consider the cases

G ≃ A_{2}(4), A_{3}(4),^{2}A_{3}(2), C_{3}(2), D_{4}(2).

For G ≃ A_{2}(4),^{2}A_{3}(2), C_{3}(2), D_{4}(2) one can check the assertion with the lists of maximal subgroups in [6]. Also A_{5}(2) may be discarded by a theorem of Kantor [21] which shows that either M ≃ GL_{3}(4).2 or M ≃ GL_{2}(8).3 has to hold.

In all the other cases Zsigmondy’s Theorem [37] implies that there exists a primitive prime divisor r|Φ_{k}^{∗}(q), Φ_{k}^{∗}(q)|ord(s) and r ≥ k + 1 (here Φ_{k}^{∗}(q) := Φ_{k}(q)/gcd(k, Φ_{k}(q))), where k is defined as follows:

type(G)  A_{l} (l even)  A_{l} (l odd)  ^{2}A_{l} (l even)  ^{2}A_{l} (l odd)  B_{l}  C_{l}  D_{l}  ^{2}D_{l}
k  l + 1  l  2l + 2  2l  2l  2l  2l − 2  2l

Let us first assume that M is not contained in C_{1} ∪ … ∪ C_{8}. Then the Main Theorem of [1] implies that H: = F^{∗}(M) = soc(M) has to be a non-abelian simple group and for some central extension H̃ of H, the natural KG̃-module V is an absolutely irreducible KH̃-module, defined over no proper subfield of K.

As r is a prime divisor of |Aut(H)| and r ≥ m_{M,p} − 1, where m_{M,p} denotes the smallest degree of a faithful projective representation in characteristic p, we claim that r||H|: Let R̃ ≤ S̃ be of order r and J̃: = ⟨ R̃^{g} | g ∈ M̃ ⟩. Then J: = ψ(J̃) \underline{◁} M and thus one h as H \underline{◁} J; in particular E(J̃) ≠ 1. Application of Theorem 2.12 of [12] allows to conclude that J̃ = E(J̃) resp. J = E(J) and the claim is proved.

Now we apply the classification of finite simple groups. Assume first that H is sporadic. Let τ = ord(s). The dimensions of faithful representations of H are bounded below [25, 5.3.8]. Since r divides |H| and r ≥ m_{H,p} − 1, this rules out immediately J_{4}, McL, HS, F i_{22}, F i_{23}, F i′_{24}, HN, Ly, T h, BM, M. In the remaining groups, τ ≤ 60, dim_{K}(V) ≥ 5, and hence all the possibilities for G are easily listed:

\begin{matrix}G ∈ \{ & A_{4}(2), A_{4}(3), A_{5}(2), A_{5}(3),^{2}A_{5}(2),^{2}A_{7}(2),^{2}A_{4}(2),^{2}A_{4}(4),^{2}A_{6}(2),^{2}A_{8}(2), B_{3}(3), B_{4}(3), \\ C_{3}(3), C_{3}(4), C_{4}(2), C_{5}(2), D_{4}(3), D_{5}(2), D_{5}(3), D_{6}(2),^{2}D_{4}(2),^{2}D_{4}(3),^{2}D_{5}(2)\}. \end{matrix}

In particular m_{H,p} ≤ 12 and hence by [25, 5.3.8] H is one of

M_{11}, M_{12}, M_{22}, M_{23}, M_{24}, J_{1}, J_{2}, J_{3}, Suz.

But it is easy to see that H ≄ Suz. Namely, m_{Suz,p} ≥ 12 [25, Prop. 5.3.8] and so G must be isomorphic to D_{6}(2) and τ = 33. But Aut(Suz) does not contain elements of order 33. So we may conclude that

τ ≤ max{ord(g)|g ∈ Aut(H), H = M_{11,...24}, J_{1,2,3}} ≤ 34.

Assume that H ≃ J_{3}. Since m_{J_{3},p} ≥ 9 ([25, Prop. 5.3.8]) the natural module for G must have dimension greater or equal to 9. This yields G ≃ D_{5}(2),^{2}D_{5}(2), D_{6}(2). But 11|\ |Aut(J_{3})|, while on the other hand 19|\ |D_{5}(2)|, therefore H ≃ J_{3} may be excluded as well. Hence we may even assume that τ ≤ 24 and that each prime divisor of τ lies in the set {2, 3, 5, 7, 11, 19, 23}. So we are left with the cases

G ∈ {^{2}A_{4}(2),^{2}A_{5}(2), B_{3}(3), C_{3}(3), D_{4}(3)}

and

H ∈ {M_{11}, M_{12}, M_{22}, M_{23}, M_{24}, J_{1}, J_{2}}.

All the maximal subgroups of the groups in the list for G are known ([6] and [23]) and this leads to the only exception M_{22} < ^{2}A_{5}(2).

Now let H be an alternating group of degree m. First consider the case where m ≤ 8. Then τ ≤ 15, all prime divisors of τ are less or equal to 7, and so all the possibilities for G can be listed:

G ∈ {A_{2}(2), A_{3}(2),^{2}A_{2}(3),^{2}A_{2}(4),^{2}A_{3}(3), C_{2}(3)}.

Inspecting the lists of maximal subgroups for these groups in [6], this leads to the embeddings A_{7} < A_{3}(2), A_{7} < ^{2}A_{2}(5), A_{7} < ^{2}A_{3}(3), S_{6} < C_{2}(3); the last case falls into family C_{3} and hence does not need to be listed separately.

Now assume that m ≥ 9. Then the dimension δ of V is at least m−2 by [25, 5.3.7]. The primitive prime divisor r of Φ_{k}(q) satisfies m ≥ r ≥ k + 1, and since also k ≥ δ − 2 we have m ≥ r ≥ δ − 1 ≥ m − 3. It follows that τ ≤ 3r. We can now easily list all the possibilities (δ ≥ 7): G is one of B_{3}(3), C_{3}(3), D_{4}(3) and D_{6}(2). The first two are covered by [6]; this leads to the embedding S_{9} < B_{3}(3). The third implies embeddings S_{9} < B_{3}(3) < D_{4}(3), but no maximal subgroup of D_{4}(3) is isomorphic to S_{9} [23]. In the last case, the only possibility is A_{14} ≤ D_{6}(2); this however does not exist: for, A_{14} has exactly one faithful representation over 𝔽_{2} of degree 12, but there is no quadratic form invariant under A_{14} (see [25, 5.3.8]).

Next we consider the case where H is a group of Lie type defined over a field of characteristic different from p. We apply the following proposition:

Proposition 1.1: Let H be a simple group of Lie type and r a prime divisor of |H|. Let δ be the degree of a faithful projective representation of H over a field of characteristic p, where p is not the natural characteristic. Assume that r ≥ δ − 1. Then one of the following has to hold:

(a) r^{2}|\ |H|, and for R ∈ Syl_{r}(H), C_{H}(R) = R and |C_{Aut(H)}(R)||2r. Furthermore r ≤ 2δ + 1 and one of

(1) H ≃ A_{1}(q), q ≠ 2, 3

- q prime  r = q  δ ≥ ^{1}/_{2}(r − 1)
- q even  r = q + 1 is prime  δ ≥ r − 2
- q odd  r = ^{1}/_{2}(q + 1) is prime  δ ≥ r − 1
- q odd  r = ^{1}/_{2}(q − 1) is prime  δ ≥ r

(2) H ≃ A_{n}(q), n ≥ 2, gcd(n + 1, q − 1) = 1

- q = 2  r = 2^{n} − 1 is prime  δ ≥ r
-  r = \frac{q^{n+1}−1}{q−1} is prime  δ ≥ ^{1}/_{2}(r − 1)

(3) H ≃ ^{2}A_{n}(q), n even, gcd(n + 1, q + 1) = 1, (n, q) ≠ (2, 2)

-  r = \frac{q^{n+1}+1}{q+1} is prime  δ ≥ r − 1

(4) H ≃ ^{2}B_{2}(8)

- r = 7, 13 δ ≥ 8

(5) H ≃ C_{n}(q), n ≥ 2

- n = 2, q = 4  r = 17  δ ≥ 18
- q = 3  r = ^{1}/_{2}(3^{n} − 1) is prime  δ ≥ r
- q odd  r = ^{1}/_{2}(q^{n} + 1) is prime  δ ≥ r − 1

(6) H ≃ G_{2}(q)

- q = 3  r = 13  δ ≥ 14
- q = 4  r = 13  δ ≥ 12

(b) H ≃ A_{1}(5), A_{1}(7), A_{1}(9), A_{2}(4),^{2}A_{3}(2),^{2}A_{3}(3), C_{3}(2), D_{4}(2) and r ≤ 7.

Proof : This is a direct consequence of the lower bounds for m_{X,π′} where X is a group of Lie type defined over a field of characteristic π obtained in [26] and the upper bound for a prime divisor of X. Explicit lists can be found in [18, Table 2]. Our list is slightly longer since we assumed r ≥ m_{X,π′} − 1. Here we used also that m_{^{2}F_{4}(2)′,2′} = 26 [19] and that any prime divisor of the order of X = ^{2}G_{2}(3^{2f+1}) has to be smaller than 3^{2f+1} + 3^{f+1} + 1.

So if case (b) of the Proposition holds we conclude that τ ≤ 30 and also that any prime divisor of τ has to be at most 7. This leaves the following cases:

G ∈ {A_{2}(2), A_{3}(2), A_{3}(3), A_{3}(4),^{2}A_{2}(3),^{2}A_{2}(5),^{2}A_{3}(3),^{2}A_{3}(5), B_{3}(3), C_{2}(3), C_{2}(7), C_{3}(3), D_{4}(3)}.

For G ≄ A_{3}(4),^{2}A_{3}(5), C_{2}(7) one can use the list of maximal subgroups in [6], [23]. This shows the embeddings A_{1}(7) < ^{2}A_{2}(3), A_{2}(4) < ^{2}A_{3}(3). The case G ≃ C_{2}(7) may be excluded because no element of order 25 is contained in Aut(H) for any group H of the list in case (b) of Proposition 1.1. For G ≃ ^{2}A_{3}(5) the only possible embedding which has to be considered is A_{2}(4).3 < ^{2}A_{3}(5). But this is ruled out by m_{A_{2}(4),5} = 6. And for G = A_{3}(4), there is no H of odd characteristic here with an element of order 21.

If on the other hand we assume that (a) holds, then τ|2r and r ≤ 2k + 1. But all groups G where τ ≤ 2(2k + 1) can be determined, giving

G ∈{A_{2}(2), A_{2}(3), A_{3}(2), A_{3}(3),^{2}A_{2}(3),^{2}A_{2}(5),^{2}A_{3}(3),^{2}A_{4}(2),^{2}A_{5}(2), B_{3}(3),
C_{2}(3), C_{3}(3), C_{4}(2), D_{4}(3), D_{5}(2),^{2}D_{4}(2)}.

This leads us to the cases A_{1}(11) ≤ ^{2}A_{4}(2) and A_{1}(17) ≤ C_{4}(2).

Now we may assume that H is a group of Lie type of the same characteristic p as V, and V is an absolutely irreducible KH̃-module of dimension δ: = dim_{K}(V). We take H to be defined over the field of p^{b} elements, and write also |K| = p^{a}.

Consider first the case where H ≃ L_{m}(p^{b}) is linear of dimension m. By [25, 5.4.6], we have b = ac for some integer c, where d = (dim_{\overline{𝔽}_{p}}(W))^{c} = dim_{K}(V) for some \overline{𝔽}_{p}H̃-module W. Now p^{xb} − 1 = p^{xac} − 1 for every x; hence we have mc ≥ k ≥ δ − 2 = (dim_{\overline{𝔽}_{p}}(W))^{c} − 2 ≥ m^{c} − 2. If a ≠ b, this implies that m = 2

and c is either 2 or 3. In the former case, we have L_{2}(p^{2a}) ≃ P Ω_{4}^{−}(p^{a}) in dimension 4 as a classical group. In the latter case, equality holds throughout the above inequality, so k = δ − 2, yielding the only possibility of L_{2}(q^{3}) in P Ω_{8}^{+}(q); however, this is not a maximal subgroup, by [23]. Hence we can assume that a = b. By [25, 5.4.11], the only possibilities for (m, δ) are (2,3), (2,4) and (4,6), both giving classical subgroups. The case (m, δ) = (2, 4) does not arise: here δ = k + 2, so the only relevant embedding would be into an orthogonal group.

Next, take H to be any other untwisted group (excluding S_{4}(2), which we have already considered). Arguing as above, [25, 5.4.6 and 5.4.13] gives a = b. Then [25, 5.4.11 and 5.4.12] implies that either the embedding of H gives a classical subgroup, or (H, δ) is one of (G_{2}, 6), p = 2; (G_{2}, 7), p ≠ 2; (S_{6}, 8), p = 2; (Ω_{7}, 8), p ≠ 2 or (P Ω_{8}^{+}, 8). The first two cases G_{2}(q) < Ω_{7}(q) resp. G_{2}(q) < Sp_{6}(q) for q even, may be excluded by some order argument. The next two cases are in conclusion (b)(ii). And the last is not of interest as the relevant embedding would be in some orthogonal group.

Now take H to be a twisted group of type A_{n}, D_{n} or E_{6}. Write σ_{0} for a graph automorphism of order 2 (or 3 if H is of type ^{3}D_{4}). First let H = U_{m}(p^{b}). By [25, 5.4.6(ii)], either a) b = ac, where δ = (dim_{\overline{𝔽}_{p}}(W))^{c} for some irreducible \overline{𝔽}_{p}H̃ module W isomorphic to W^{σ_{0}}, or b) 2b = ac, a does not divide

b, δ = (dim_{\overline{𝔽}_{p}}(W))^{c} for some \overline{𝔽}_{p}H̃-module W not isomorphic to W^{σ_{0}}. Moreover, it is easy to tell whether or not W is isomorphic to W^{σ_{0}} - see [25, page 192]. In case a), since p^{2xb} = p^{2xac} for every x, we have 2mc ≥ k ≥ δ − 2 ≥ (dim_{\overline{𝔽}_{p}}(W))^{c} − 2. Now dim_{\overline{𝔽}_{p}}(W) ≥ 6 and in fact dim_{\overline{𝔽}_{p}}(W) ≥ m(m − 1) for m ≥ 7 by [25, 5.4.8]. Hence c = 1, a = b and m ≤ 6. Moreover, [25, 5.4.8] now forces m to be 3 or 4, and so δ is 6, 7 or 8. If m = 4, [25, 5.4.11] shows that the only relevant embedding is classical in dimension 6. If m = 3, a similar argument shows that the only embedding of interest here is in dimension 8 if p ≠ 3 and in dimension 7 for p = 3; in particular G = P Ω_{8}^{+}(q) and M ≃ P GU_{3}(q) (cf. [23]) or G = Ω_{7}(q) and M ≃ U_{3}(q) if 3|q. But P GU_{3}(q) does not contain a cyclic subgroup of order ^{1}/_{d} (q^{3} + 1) and thus this embedding is not possible. Also the second case may be discarded as no maximal subgroup of Ω_{7}(q) is isomorphic to U_{3}(q) [24]. In case b), if c ≥ 2 then c ≥ 3; now p^{2xb} = p^{xac}, so mc ≥ k ≥ δ − 2 ≥ m^{c} − 2, which is impossible. So c = 1 and hence δ is less than 2m, and [25, 5.4.11] shows that there are no examples. A similar but easier proof applies to the other groups here. For ^{3}D_{4}(p^{b}), we use 5.4.7(a), 5.4.8 and 5.4.11 of [25]. Also, ^{2}E_{6}(p^{b}) is ruled out by [25, 5.4.6, 5.4.8 and 5.4.13]. Finally, the only possibility for P Ω_{m}^{−}(p^{b}) is the classical embedding.

Finally we consider the groups ^{2}B_{2}(2^{b}),^{2}G_{2}(3^{b}),^{2}F_{4}(2^{b}) in characteristic 2, 3, 2, respectively. Here we use [25, 5.4.7(b)]. There are no examples for ^{2}F_{4}. The only possibilities for the other two families are in dimensions 4 and 7, respectively. However, in neither case does H contain an element of the right order (note that ^{2}B_{2}(2) is excluded here).

Thus we may assume that M is contained in C_{1} ∪ … ∪ C_{8}. Now let us consider M̃ instead of M (M = ψ(M̃)). There exists a primitive prime divisor r of Φ_{k}^{∗}(q), where r ≥ k+1, and for R ∈ Syl_{r}(S̃), W ≤ V is an irreducible KR-module with dim_{K}(W) > ^{1}/_{2} dim_{K}(V). So it follows that M ∉ C_{4} ∪ C_{5} ∪ C_{7}. Assume that M ∈ C_{2}. Then M̃ is an imprimitive subgroup of GL_{K}(V) stabilizing a decomposition V = V_{1} ⊕⋯⊕V_{m}. As r ≥ k + 1 and R ⊈ StabM̃(V_{1}), one obtains dim_{K}(V_{i}) = 1, and G has to be of type B_{l} or D_{l}. Then the V_{i}’s must be anisotropic subspaces and therefore s̃ ∈ CM̃(R) implies ord(s̃)|2r and r ≤ dim_{K}(V). This cannot happen in our case, hence also M ∉ C_{2}.

Let us assume that M̃ ≥ S̃ is contained in C_{6}. Then 4.6 of [25] implies that r ≤ dim_{K}(V) + 1. As r ≥ dim_{K}(V) − 1 one gets that R ∈ Syl_{r}(M̃) has order r and CM̃(R) = Z(G̃).R. This implies that ord(s) = τ = k + 1 and that G is not of type ^{2}A_{l}. But all the possibilities for G where τ = k + 1 can be listed and this leads to G ≃ C_{2}(2) or G ≃ C_{2}(3). The first case was excluded, while the second one leads to M = 2^{4}.A_{5} < C_{2}(3). Apart from that one gets M̃ ∉ C_{6}.

If G is of type A_{l} or ^{2}A_{l} with l even, C_{l} or ^{2}D_{l}, V is an irreducible KM̃-module and therefore M̃ ∉ C_{1}. If G is of type A_{l} one has |S̃| = \frac{q^{l}−1}{q−1} and there are exactly 2 non conjugate maximal parabolic subgroups P_{1}, P_{2} ∈ C_{1} containing S̃. If G ≃ ^{2}A_{l}(q) with l odd, we get

S̃ < SU_{l+1} ∩ (GU_{1}(q) ⊥ GU_{l}(q)) = M̃ < SU_{l+1}(q) = G̃.

There is only one maximal subgroup M̃ ∈ C_{1} containing S̃. If G ≃ B_{l}(q) for q odd, one gets

S̃ < Ω_{2l}^{−}(q).2 = M̃ < Ω_{2l+1}(q) = G̃.

There is also only one maximal subgroup M̃ ∈ C_{1} containing S̃. If G ≃ D_{l}(q) one gets two types of embeddings

S̃ < Ω_{2l}^{+}(q) ∩ (GO_{2}^{−}(q) ⊥ GO_{2l−2}^{−}(q)) = M̃_{0} < Ω_{2l}^{+}(q) = G̃

and

S̃ < Ω_{2l}^{+}(q) ∩ (GO_{1}(q) ⊥ GO_{2l−1}(q)) = M̃_{1} < Ω_{2l}^{+}(q) = G̃.

There is only one maximal subgroup isomorphic to M̃_{0} containing S̃, but there are (q +1) GO_{2l}^{+}(q)-conjugacy classes of maximal subgroups isomorphic to M̃_{1} containing S̃.

If M̃ ∈ C_{8}, G has to be of type A_{l} or C_{l} ([25, 4.8.A]). But for G ≃ A_{l} an easy argument shows that S̃ ≤ M̃ implies M̃ ∉ C_{8}. For G ≃ C_{l}(q) with q even, we obtain the embedding

S̃ < Ω_{2l}^{−}(q).2 = M̃ < Sp_{2l}(q).

If l ≠ 2 there is only one maximal subgroup M̃ ≃ Ω_{2l}^{−}(q).2 containing S̃, if l = 2 there are 2, conjugate under the graph automorphism of Sp_{4}(q) [35].

Thus we may assume that M̃ ∈ C_{3}. Then Table 4.3.A of [25] implies that if M̃ is a maximal subgroup of G̃ containing S̃, M̃ has to be one of the groups described in Table 1.2.

Table 1.2

type(G)  G̃  M̃  conditions
A_{l}  SL_{l+1}(q)  SL_{l+1}(q) ∩ GL_{l+1/k}(q^{k}).k  l even, k|l + 1
2A_{l}  SU_{l+1}(q)  SU_{l+1}(q) ∩ GU_{l+1/k}(q^{k}).k  l even, k|l + 1
C_{l}  Sp_{2l}(q)  Sp_{2l/k}(q^{k}).k  k|l
GU_{l}(q).2  l odd, q odd
D_{l}  Ω_{2l}^{+}(q)  Ω_{2l}^{+}(q) ∩ GO_{l}(q^{2}).2  l odd
Ω_{2l}^{+}(q) ∩ GU_{l}(q).2  l even
2D_{l}  Ω_{2l}^{−}(q)  Ω_{2l}^{−}(q) ∩ GO_{2l/k}^{−}(q^{k})
Ω_{2l}^{−}(q) ∩ GU_{l}(q).2  l odd

This completes the proof of Theorem 1.1.

As an immediate consequence of Theorem 1.1. one gets that the set

Gen_{G}(s) = {g ∈ G|⟨s, g⟩ = G}

comprises, in a sense, nearly the whole group G (cf. [36, Theorem A]). This is made precise in the next theorem.

Theorem 1.2:

(a) Let G be a finite simple classical group of type A_{l}, ^{2}A_{l} for l even, or ^{2}D_{l}, with field of definition 𝔽_{q} . Let s ∈ G be as before. Then for each ε > ε_{type(G)} there exists a constant C such that for q ≥ C,

|Gen_{G}(s)| ≥ |G| − |G|^{ε},

and the values for ε_{type(G)} are as follows:

\begin{matrix}type(G) & A_{l} & ^{2}A_{l} (l even) & ^{2}D_{l} \\ ε_{type(G)} & min\{k; k|(l + 1), k ≠ 1\}^{−1} & min\{k; k|(l + 1), k ≠ 1\}^{−1} & ^{4}/_{7} \end{matrix}

(b) Let G be a finite simple classical group of type ^{2}A_{l} for l odd, B_{l}, C_{l}, or D_{l}, with field of definition 𝔽_{q}. Let s ∈ G be as before. Then for each ϑ < ϑ_{type(G)} there exists a constant C such that for q ≥ C,

|Gen_{G}(s)| ≥ |G| (1 − \frac{1}{q^{ϑ}} ) ,

and the values for ϑ_{type(G)} are as follows:

\begin{matrix}type(G) & ^{2}A_{l} (l odd) & B_{l} & C_{l} & D_{l} \\ ϑ_{type(G)} & 2l & 2l & 2l & 2l − 2 \end{matrix}

Proof : We estimate the order of the set X = G \ Gen_{G}(s). So

X = ⋃_{s∈M<G} M.

For G ≃ A_{l}(q) we obtain from Theorem 1.1 and Table 1.3:

|X| ≤ \frac{1}{d · (q − 1)} ∑_{{k|(l+1) \atop k≠1}} k · |GL_{l+1/k}(q^{k})|

≤ ∑_{{k|(l+1) \atop k≠1}} k · |G|^{1/k} + o(|G|^{1/ρ})

≤ σ(l + 1) · |G|^{1/ρ} + o(|G|^{1/ρ})

where ρ denotes the smallest divisor of (l + 1) not equal to 1. So the assertion holds for groups of type A_{l}. For the classical groups of other types the estimates are similar and also straightforward.

Finally we want to mention that a similar result holds for the set Gen_{G}^{2} = {(g, h) ∈ G × G|⟨g, h⟩ = G} which was proved by W. Kantor and A. Lubotzky [22]. They also used the information on maximal subgroups of the classical groups to show that |Gen_{G}^{2}| /|G|^{2}⟶ 1 as |G| ⟶ ∞.

2 Three involutions

In this section we investigate certain structure constants of the finite classical groups and the groups of type E_{6}. The strategy is to show that these do not vanish for certain triples of classes. These triples are so chosen that in the next section we can combine this information with the results of section 1 to obtain our theorems. The calculation of the structure constants makes use of the Deligne-Lusztig theory of characters of reductive groups.

Namely let C_{i}, i = 1, 2, 3, be conjugacy classes of a finite group S with trivial center. Define

Σ̄(C_{1}, C_{2}, C_{3}) := {(σ_{1}, σ_{2}, σ_{3}) | σ_{i} ∈ C_{i}, σ_{1}σ_{2}σ_{3} = 1}.

It is well known that the cardinality of the set Σ̄(C_{1}, C_{2}, C_{3}) can be computed from a knowledge of the irreducible complex characters of S, using that the center Z(S) is trivial:

Lemma 2.1: Let χ_{1}, …, χ_{h} be the irreducible complex characters of S. Then

\frac{|Σ̄(C_{1}, C_{2}, C_{3})|}{|S|} = \frac{|S|}{|C_{S}(σ_{1})||C_{S}(σ_{2})||C_{S}(σ_{3})|} ∑_{i=1}^{h} \frac{χ_{i}(σ_{1})χ_{i}(σ_{2})χ_{i}(σ_{3})}{χ_{i}(1)} =: n(C_{1}, C_{2}, C_{3}).

We call n(C_{1}, C_{2}, C_{3}) the normalized structure constant of (C_{1}, C_{2}, C_{3}). If we can show that for a suitable choice of C_{i} – if one class is a certain class of strongly real elements and one class is a certain class of involutions – one has n(C_{1}, C_{2}, C_{3}) > 0, and moreover that any pair of elements σ_{1} ∈ C_{1}, σ_{2} ∈ C_{2} must generate S, then the above generation property is proved.

In the following we will always choose C_{1} and C_{2} to contain regular elements of tori T_{1}, T_{2}. For the convenience of the reader we collect the orders of the various T_{i} in the following table:

Table 2.1: The tori T_{1} and T_{2}.

|T_{1}|  |T_{2}|
A_{l}  (q^{l+1} − 1)/(q − 1)  q^{l} − 1
2A_{l}  (q^{l+1} − (−1)^{l+1})/(q + 1)  q^{l} − (−1)^{l}
B_{l}  q^{l} + 1  q^{l} − 1
C_{l}  q^{l} + 1  q^{l} − 1
D_{l} (l odd)  q^{l} − 1  (q^{l−1} + 1)(q + 1)
D_{l} (l even)  (q^{l−1} + 1)(q + 1)  (q^{l/2} + (−1)^{l/2})^{2}
2D_{l}  q^{l} + 1  (q^{l−1} + 1)(q − 1)

2.1. The cases A_{l} and ^{2}A_{l}

Let G = P GL_{l+1}(q) be the adjoint group of type A_{l} over the field with q = p^{n} elements. Then G has maximal tori T_{1}, T_{2} of orders (q^{l+1} − 1)/(q − 1) and q^{l} − 1 respectively. As gcd(q^{l+1} − 1, q^{l} − 1) = q − 1 we get gcd(|T_{1}|, |T_{2}|) = gcd(l + 1, q − 1) =: d. Hence the subgroups T′_{i} := T_{i} ∩ G′ in G′ = P SL_{l+1}(q) of orders |T′_{i}| = |T_{i}|/d have coprime orders. Now the values of the irreducible characters of G on semisimple elements may be calculated from the values of the Deligne-Lusztig characters (see [4]). The Deligne-Lusztig characters R_{T,θ} are parametrized by pairs (T, θ). The equivalence relation of geometric conjugacy on these pairs defines a partition of the irreducible characters of G into disjoint series as follows: The geometric conjugacy classes of pairs (T, θ) can be parametrized by conjugacy classes (s) of semisimple elements in the dual group Ĝ = SL_{l+1}(q). Define E(s) to be the set of irreducible characters χ occurring as a constituent in one of the R_{T,θ} with (T, θ) associated to (s). Then the sets E(s), where (s) runs over the set of semisimple conjugacy classes of Ĝ, constitute a partition of Irr(G) (see e.g. [4, 7.3.8 and 7.5.8]).

Now the R_{T,θ} span the space of class functions restricted to semisimple elements ([4, 7.5.7]). In particular, R_{T,θ} = R(s) vanishes on regular elements of a torus T′ if the element s is not conjugate in Ĝ to an element in the dual T̂′ of T′. So to determine the R_{T,θ} not vanishing on regular elements of T_{1} and T_{2}, we just have to consider conjugacy classes (s) of elements s ∈ Ĝ which lie in the intersection of the dual tori T̂_{1} and T̂_{2}. It is well known that |T̂| = |T | [4, 4.4.2], so by the above remark about gcd(|T_{1}|, |T_{2}|) it follows that all relevant elements s lie in the center of SL_{l+1}(q) of order d.

Hence only those R(s) with s central in Ĝ = SL_{l+1}(q) may possibly not vanish on regular elements of both tori T_{1} and T_{2}. It is known that the irreducible constituents of such R(s) are precisely the extensions of the unipotent characters of P SL_{l+1}(q) to P GL_{l+1}(q). (Since the unipotent characters of SL_{l+1}(q) have the center in their kernel, they can be regarded as irreducible characters of P SL_{l+1}(q), thus defining the term unipotent character for P SL_{l+1}(q).) Summing up we may state:

Proposition 2.1: The irreducible characters of P SL_{l+1}(q) not vanishing on regular elements of the two maximal tori of orders (q^{l+1} −1)/(d(q −1)) and (q^{l} −1)/d are among the unipotent characters of P SL_{l+1}(q).

Now let C_{i}, i = 1, 2, be conjugacy classes of some regular elements of G′ in T′_{i} with the following properties: if l is odd, then elements in C_{1} have order (q^{(l+1)/2} + 1)/gcd(2, q − 1) and those in C_{2} have order (q^{l} − 1)/d, if l is even, then those in C_{1} have order (q^{l+1} − 1)/(d(q − 1)) and those in C_{2} have order q^{l/2} + 1. Further, let C_{3} be any non-identity class of G′. Using the above lemma and the observations about the characters of G′ not vanishing on C_{1} and C_{2} we obtain:

Theorem 2.1: (a) The normalized structure constant n(C_{1}, C_{2}, C_{3}) in G′ = P SL_{l+1}(q) is non-zero. (b) The classes C_{1} and C_{2} are non-empty for each l ≥ 1 with (l, q) ≠ (1, 2), (1, 3).

(c) If l is odd, the elements in C_{1} are strongly real; if l is even, the elements in C_{2} are strongly real.

Proof : (a) We use Lemma 1 to calculate n(C_{1}, C_{2}, C_{3}). As only those characters not vanishing on both C_{1} and C_{2} will contribute to the sum, by Proposition 1 it suffices to treat the unipotent characters ρ_{i}. The degrees of these characters, as well as the order of G, can be viewed as polynomials in q, and, more precisely, products of cyclotomic polynomials in q times a power of q. By a well known vanishing result for character values, if the degree of an irreducible character of a group H is divisible by the entire r-part of |H|, for r a prime dividing |H|, then this character vanishes on all r-singular elements of H. In our case a unipotent character ρ_{i} takes the same value on all regular elements of a given maximal torus T̄ of G′. Let r be a primitive prime divisor of |T_{1}| in the sense of Zsigmondy [37]. Then by definition the r-part of (q^{l+1} − 1)/(q − 1) is the same as the r-part of |G|, as the l + 1-th cyclotomic polynomial Φ_{l+1}(q) divides |G| just once. Hence any character with degree divisible by Φ_{l+1}(q) vanishes on all r-singular elements, hence on all regular elements of T_{1}. In the exceptional case l = 1 of Zsigmondy’s theorem, the character table is completely known and the above assertion may be checked directly; for (l, q) = (5, 2) it may be deduced from the behavior for (5, q), q > 2, since the character values on regular semisimple elements are generic, i.e., independent of q. The same consideration applies to characters with degree divisible by Φ_{l}(q), which have to vanish on all regular elements of T_{2}.

The unipotent characters ρ of G are parametrized by partitions λ of l+1 (see for example [4, 13.8]). The degrees of the ρ_{λ} may also be found in [4, 13.8]. We see that Φ_{l+1}(q) does not divide the degree ρ_{λ}(1) only if the Young diagram for λ contains an l + 1-hook. Also, Φ_{l}(q) does not divide ρ_{λ}(1) if the Young diagram contains an l-hook. But the only partitions of l + 1 whose diagrams contain both an l- and an l + 1-hook are (l + 1) and (1)^{l+1}. These correspond to the trivial and the Steinberg character St of G. So we may calculate

n(C_{1}, C_{2}, C_{3}) = \frac{|G′|}{|C_{G′}(σ_{1})||C_{G′}(σ_{2})||C_{G′}(σ_{3})|} (1 + \frac{St(σ_{1})St(σ_{2})St(σ_{3})}{St(1)}) .

As σ_{1} ∈ C_{1} and σ_{2} ∈ C_{2} are regular, |St(σ_{i})| = 1 for i = 1, 2 by [4, 6.4.7]. Finally, |St(σ_{3})| is smaller than St(1) = |G′|_{p} except if σ_{3} ∈ Z(G′). But the center Z(G′) = Z(P SL_{l+1}(q)) is trivial, so we get

(1 + \frac{St(σ_{1})St(σ_{2})St(σ_{3})}{St(1)}) > 0

as desired.

(b) The torus T_{1} is cyclic, being a homomorphic image of the torus of GL_{l+1}(q) generated by a Singer cycle. Let x be an element in T′_{1} whose order is divisible by a primitive prime divisor of Φ_{l+1}(q) in the sense of Zsigmondy. Then clearly this element cannot lie in any other torus, so it is regular and C_{1} is non-empty. In the exceptional case l = 1 we have q ≥ 4, because we are interested in nonsolvable groups only. It is well known that both types of tori of P SL_{2}(q) contain regular elements for q ≥ 4. If on the other hand (l, q) = (5, 2), we have G′ = P SL_{6}(2), and by [15] there exist 54 regular elements in the torus T_{1} of order 63; in particular elements of order 9 in G′ lie in C_{1} (the non-regular elements having orders dividing 3 or 7). Basically the same arguments apply to the class C_{2}. The exceptions here occur for P SL_{3}(q) (but there it is known that all elements of order q + 1 are regular, see for example [15]), and for P SL_{7}(2) the torus T_{2} again contains 54 regular elements by [15].

(c) We use the characterization of strongly real by containment in a dihedral subgroup. If l is odd, then P SL_{l+1}(q) contains a subgroup P SL_{2}(q^{(l+1)/2}) and hence a dihedral group of order gcd(2, q) (q^{(l+1)/2} + 1). If l is even, we use the fact that GL_{l}(q) < SL_{l+1}(q): for, GL_{2}(q^{l/2}) < GL_{l}(q), and GL_{2}(q^{l/2}) contains an involution inverting a cycle of order q^{l/2} + 1.

Now let G = P GU_{l+1}(q), the projective general unitary group. Choose T_{1}, T_{2} to be maximal tori of orders (q^{l+1} − (−1)^{l+1})/(q + 1) and q^{l} − (−1)^{l}. Then these are the Ennola duals of the tori chosen for type A_{l}. Let C_{i}, i = 1, 2, be classes of regular elements in T′_{i} := T_{i} ∩ G′ with G′ = P SU_{l+1}(q), of orders (q^{(l+1)/2} + (−1)^{(l+1)/2})/gcd(2, q − 1), resp. (q^{l} + 1)/d if l is odd, (q^{l+1} + 1)/(d(q + 1)), resp. q^{l/2} + (−1)^{l/2} if l is even, and C_{3} a class of non-identity elements in G′.

Theorem 2.2: (a) The normalized structure constant n(C_{1}, C_{2}, C_{3}) in G′ = P SU_{l+1}(q) is non-zero. (b) The classes C_{1} and C_{2} are non-empty for each l ≥ 2 with (l, q) ≠ (2, 2), (2, 3).

(c) If l is odd, the elements in C_{1} are strongly real; if l is even, the elements in C_{2} are strongly real.

Proof : (a) The first part follows from the previous proof by the Ennola duality for character values of linear and unitary groups, which was proved by Lusztig and Srinivasan [27].

(b) By [16, 3.3 and 6.7], the tori T_{1} and T_{2} are cyclic. We may again quote Zsigmondy to obtain the existence of regular elements in the two tori. In the exceptional case (l, q) = (5, 2) there exist 18 regular elements in T_{1} (for example we may take C_{1} = 7A in Atlas notation [6]), for (l, q) = (6, 2) the list in [15] shows 54 regular elements in T_{2}, which are precisely those of order not dividing 9. If l = 2, hence G′ = P SU_{3}(q), the class C_{2} may be taken to contain elements of order q − 1 > 2 unless q = 2 (when G′ is solvable) or q = 3 (which was excluded).

(c) In the case of odd l we have SL_{2}(q^{(l+1)/2}) < SU_{l+1}(q), and an argument as in the previous proof shows the assertion. For l even, the embedding GU_{l}(q) < SU_{l+1}(q) serves the same purpose as the corresponding embedding in the linear case, completing the proof.

2.2. The cases B_{l} and C_{l}

Now let G = Sp_{2l}(q), the simply connected group of type C_{l} over the field with q elements. Denote by T_{1}, T_{2} maximal tori of G of orders q^{l} + 1, q^{l} − 1 respectively. Clearly gcd(|T_{1}|, |T_{2}|) = (q − 1, 2) =: d. Hence in Ḡ = S_{2l}(q) = G/Z(G) of order |G|/d, the corresponding subgroups T̄_{1} and T̄_{2} have coprime orders. So again we are in a situation as in the first part, and by the same reasoning the only irreducible characters not simultaneously vanishing on regular elements of both tori are among the unipotent characters of Ḡ. Let again C_{i}, i = 1, 2, consist of regular elements of T̄_{i}, and C_{3} be any class of Ḡ containing non-central elements.

Before we state the main result of this section, we first introduce the notion of a symbol. These are important because the unipotent characters of groups of classical type other than A_{l} are parametrized by such symbols. A symbol is a pair of sequences of strictly increasing non-negative integers, usually denoted as

Λ = ( \begin{matrix}λ_{1} & … & λ_{a} \\ μ_{1} & … & μ_{b} \end{matrix} ) .

Two symbols are considered to be equivalent if they can be transformed into each other by a sequence of shifts

( \begin{matrix}λ_{1} & … & λ_{a} \\ μ_{1} & … & μ_{b} \end{matrix} ) ∼ ( \begin{matrix}0 & λ_{1} + 1 & … & λ_{a} + 1 \\ 0 & μ_{1} + 1 & … & μ_{b} + 1 \end{matrix} ) ,

or interchanging the two rows. The rank of a symbol Λ is rank(Λ) := ∑ λ_{i} + ∑ μ_{i} − [(\frac{a+b−1}{2})^{2}], and d := |a − b| is called the defect of Λ (there should be no confusion with d = gcd(q − 1, 2)).

In a similar fashion as for partitions, r-hooks of such symbols can be defined. First, let r be an odd positive integer. Then an r-hook of a symbol Λ is a pair (a_{1}, a_{2}) of non-negative integers with a_{2} = a_{1} − r, such that either a_{1} ∈ {λ_{i}} and a_{2} ∉ {λ_{i}}, or a_{1} ∈ {μ_{i}} and a_{2} ∉ {μ_{i}}. To remove an r-hook now means to replace the occurrence of a_{1} in the first row (resp. second row) by a_{2} in the same row (and then reorder the entries). Clearly this leads to a symbol with rank reduced by r.

If r = 2e is an even positive integer, then an r-hook is again a pair (a_{1}, a_{2}), with now a_{2} = a_{1} − e, and either a_{1} ∈ {λ_{i}} and a_{2} ∉ {μ_{i}}, or vice versa. To remove such an r-hook means to delete the entry a_{1} in the first (resp. second) row and to insert an entry a_{2} in the second (resp. first) row at the appropriate place. This process reduces the rank by e = r/2.

Theorem 2.3: (a) The normalized structure constant n(C_{1}, C_{2}, C_{3}) in Ḡ = S_{2l}(q) is non-zero. (b) The classes C_{1} and C_{2} are non-empty for each l ≥ 2 with (l, q) ≠ (2, 2).

(c) The elements in C_{1} are strongly real.

Proof : (a) We follow the ideas in the proof of Theorem 1. Namely, any character with degree divisible by either Φ_{2l}(q) or Φ_{l}(q) will vanish on either C_{1} or C_{2} and hence not contribute to the sum for the normalized structure constant.

Lusztig has shown that the unipotent characters of groups of type C_{l} are parametrized by symbols of rank l and odd defect (see for example [4, 13.8]). The formulae for the degrees of unipotent characters ρ_{Λ} may again be found in [4, 13.8]. In [33] it was shown that the degree ρ_{Λ}(1) is not divisible by Φ_{2l}(q) if and only if a 2l-hook can be removed from Λ, and also it is not divisible by Φ_{l}(q) if and only if an l-hook can be removed. So we have to study symbols of rank l and odd defect from which a 2l- and an l-hook can be removed.

As 2l is even, removing a 2l-hook reduces the rank of Λ by l. The resulting symbol hence will have rank 0. The symbols of rank zero clearly have the form

Λ′ = ( \begin{matrix}0 & 1 & … & & a \\ 0 & 1 & … & a − 1 \end{matrix} ) .

By shifting the entries of Λ′ to the left or to the right we see that we may assume without loss of generality that the removed hook had the form (l, 0). There are two possibilities for Λ, depending on whether the hook was removed in the top or in the bottom row. We arrive at

Λ = \binom{1 … a}{0 1 … a − 1 l}

or

Λ = \binom{0 1 … a l}{1 … a − 1} .

In the first case clearly l > a − 1, so 0 ≤ a ≤ l, and this leads to l + 1 different symbols up to equivalence. In the second case, l > a, so 1 ≤ a ≤ l − 1, and this gives l − 1 possibilities. In total, there remain 2l unipotent characters of G to consider.

Next, those of the remaining symbols containing an l-hook have to be found. Here we distinguish two cases, according to whether l is even or odd. First assume l odd. Then the removal of an l-hook also reduces the rank by l, and it is easy to see that of the above symbols only those with a = 0 or a = l remain in the first case, i.e.,

Λ_{1} = ( \begin{matrix}l \\ − \end{matrix} ) ,  Λ_{2} = \binom{1  … l}{0 1 … l − 1 l} ,

and the second case leads to a contradiction. The symbols Λ_{1} and Λ_{2} belong to the trivial and to the Steinberg character of G, and now arguments as in the proof of Theorem 1 yield the result, as again C_{1} and C_{2} contain regular elements.

If l is even, more possibilities arise. Namely we get

Λ = \binom{1  … a}{0 1 … a − 1 l} ,  0 ≤ a ≤ l, a ≠ l/2

and

Λ = \binom{0 1 … a l}{1 … a − 1} , 1 ≤ a ≤ l − 1.

In all, 2l − 1 symbols contain 2l- and l-hooks. Hence some more considerations are necessary. For this we study the division of these characters into families, as defined for example in [4, 13.8]. If Λ is a symbol, then all other valid symbols obtainable by moving some entries from one row to the other row and vice versa belong to the same family as Λ. By the structure of the remaining symbols, they all lie in families with at most four members. More precisely,

\binom{0 … a − 1 l}{1 … a} , \binom{1 … a l}{0 … a − 1} , ( \begin{matrix}0 & … & & a − 1 & a \\ 1 & … & a − 1 & l \end{matrix} ) , \binom{0  …  a l}{1 … a − 1}

form a family, for all 1 ≤ a ≤ l − 1, and all remaining symbols apart from

( \begin{matrix}l \\ − \end{matrix} ) and \binom{1  … l}{0 1 … l − 1 l}

occur in one of these families. To calculate the values of the corresponding unipotent characters on the regular elements in T_{1} and T_{2}, one first has to determine the values of the almost characters belonging to these families and then multiply the vector of these values by the Fourier transform matrix for the family, see [4, 13.8]. The defect of the first three characters in each family is obviously equal to one, while the last one has defect three. So there are three almost characters R_{χ} associated to each family, indexed by irreducible characters χ of the Weyl group of C_{l}. The character χ is obtained by interpreting the two rows of a symbol as a pair of beta-sets for partitions, so giving rise to a pair of partitions (λ_{1}, λ_{2}) of l_{1} + l_{2} = l. In our examples, the relevant pairs of partitions are

(l − a, 1^{a}), (1^{a}(l − a), −), (−, 1^{a−1}(l − a + 1))

(the fourth symbol has defect greater than 1, hence does not lead to an almost character). The values of the almost characters R_{χ} on the regular elements of the tori T_{1} and T_{2} are by definition of the almost characters and by the character formula for the Deligne-Lusztig characters R_{T,1} equal to the values of the irreducible characters χ of W (C_{l}) on the elements of types (−, l) and (l, −) of W (C_{l}). By [20, 2.3.17], we get

(l − a, 1^{a})  (1^{a}(l − a), −)  (−, 1^{a−1}(l − a + 1))
(−, l)  0  (−1)^{a}  (−1)^{a}
(l, −)  0  (−1)^{a}  (−1)^{a−1}

as the values of the characters χ on the elements of types (−, l) and (l, −). If we apply the Fourier transform matrix for four-element families [4, 13.6] to these values, we get that the resulting unipotent characters all either vanish on regular elements of T_{1} or of T_{2}. Hence again all characters apart from the trivial and the Steinberg character do not contribute to the normalized structure constant, and the assertion follows as in the first case.

(b) The two tori T_{1} and T_{2} are cyclic by [16, 4.5]. The existence of regular elements in both tori follows immediately from Zsigmondy’s theorem. In the case l = 2, G = Sp_{4}(q), we may use the list of regular conjugacy classes in [15] to check the assertion if (l, q) ≠ (2, 2), as well as for Sp_{6}(2) and Sp_{12}(2).

(c) The Weyl group of type C_{l} contains −1, so all centralizers of elements in W (C_{l}) have even order. In particular, N_{G}(T_{1})/T_{1} contains the involution −1 which inverts every element in the torus T_{1}. This may be lifted to an element of order at most four in the normalizer as can be seen by embedding T_{1} into Sp_{2}(q^{l}) < Sp_{2l}(q). The square lies in T_{1} and has order at most two. If |T_{1}| is odd (i.e., q is even), we are done as in the case of type A, since then there exists a dihedral group in N_{G}(T_{1}) containing elements from C_{1}. If |T_{1}| is even, then the unique element of order two in the cyclic torus T_{1} lies in Z(G) and so maps to 1 in Ḡ. So in Ḡ we have again found a dihedral group containing elements from C_{1}.

Choosing the ‘same’ maximal tori for the adjoint groups of type B_{l} for odd q we obtain:

Theorem 2.4: (a) The normalized structure constant n(C_{1}, C_{2}, C_{3}) in G′ = O_{2l+1}(q) is non-zero. (b) The classes C_{1} and C_{2} are non-empty for each l ≥ 3 and 2 |̸ q.

(c) The elements in C_{1} are strongly real.

Proof : The first part is obvious, because the classification of unipotent characters for type B_{l} is exactly the same as for type C_{l}.

(b) The two tori are again cyclic by [16, 4.5]. The exceptional cases of Zsigmondy’s theorem do not occur here since l ≥ 3 and q ≠ 2.

(c) Observe that GO_{2}^{−}(q^{l}) < GO_{2l}^{−}(q) < SO_{2l+1}(q). Now GO_{2}^{−}(q^{l}) is dihedral of order 2(q^{l} + 1). Hence O_{2l+1}(q) contains a cyclic or a dihedral subgroup of order q^{l} + 1. It does not contain the former, so it must contain the latter.

2.3. The case ^{2}D_{l}

In this section, we take G = SO_{2l}^{−}(q) to be the special orthogonal group of type ^{2}D_{l} over the field with q = p^{n} elements. Let T_{1}, T_{2}, denote maximal tori of orders q^{l} + 1, (q^{l−1} + 1)(q − 1) respectively. Here gcd(|T_{1}|, |T_{2}|) = gcd(q^{l} + 1, 4) =: d. We define T′_{i} := T_{i} ∩ G′ with the normal subgroup G′ = Ω_{2l}^{−}(q) of index 2. Let C_{1} contain regular elements of the torus T′_{1}, let C_{2} be a class of generators of the cyclic subtorus of T′_{2} obtained by intersecting T_{2} with SO_{2l−1}(q), and let C_{3} be any non-central class of G′. Then with the factor group Ḡ := O_{2l}^{−}(q) of G′ by its center and the images C̄_{i} of C_{i} in Ḡ we have:

Theorem 2.5: (a) The normalized structure constant n(C̄_{1}, C̄_{2}, C̄_{3}) in Ḡ = O_{2l}^{−}(q) is non-zero. (b) The classes C̄_{1} and C̄_{2} are non-empty for each l ≥ 4 with (l, q) ≠ (4, 2).

(c) The elements in C̄_{2} are strongly real.

Proof : (a) The proof follows those of the previous four theorems. We first check the non-vanishing of the structure constant. By the remark about the orders of the tori T_{i} we again conclude that only unipotent characters may contribute to the normalized structure constant in Ḡ. For groups of type ^{2}D_{l}, the unipotent characters are parametrized by symbols of rank l and defect congruent 2 (mod 4) [4, 13.8]. As in the proof of Theorem 3 we have to classify those symbols which contain a 2l- and a 2l − 2-hook, because |T_{1}| contains a primitive prime divisor of Φ_{2l}(q) and |T_{2}| one of Φ_{2l−2}(q).

Similar considerations as in the preceding proof show that the first condition forces

Λ = \binom{0 1 … a l}{1 … a}

for 0 ≤ a ≤ l − 1, which leads to l different cases. Of these, only a = 0 or a = l − 1, i.e.,

Λ = \binom{0 l}{−} and Λ = \binom{0 … l − 1 l}{1 … l − 1}

allow the removal of a 2l−2-hook. Once more, these correspond to the trivial and to the Steinberg character, and by an argument as in the previous proofs we reach the conclusion.

(b) The existence of regular elements in the two tori is clear except for the Zsigmondy case (l, q) = (4, 2). The list of regular classes in [15] shows that indeed no regular elements exist in the torus T′_{2} of Ω_{8}^{−}(2).

(c) The assertion about elements in C_{2} (and hence in C̄_{2}) is now clear from Theorem 4 (resp. Theorem 3 if q is even), since these elements already lie in the dihedral subgroup of the normalizer of T_{1} for type B_{l−1} and hence strongly real.

2.4. The case D_{l}, l odd

As an immediate corollary to the result of the last section, we may treat the case of the group G = SO_{2l}^{+}(q) of type D_{l} over the field with q elements, when l is odd. Then D_{l} is the Ennola dual of ^{2}D_{l}, and by the results of Lusztig, the parametrization of unipotent characters for types D_{l} and ^{2}D_{l} is dual in that sense. Hence if we take T_{1}, T_{2} to be two maximal tori of G of orders q^{l} − 1, (q^{l−1} + 1)(q + 1) respectively, and C_{i}, i = 1, 2, classes of regular elements in the tori T′_{i} in G′ as above, C_{3} any other class of non-central elements, then with Ḡ := O_{2l}^{+}(q) we get:

Theorem 2.6: (a) The normalized structure constant n(C̄_{1}, C̄_{2}, C̄_{3}) in Ḡ = O_{2l}^{+}(q), l odd, is non-zero. (b) The classes C̄_{1} and C̄_{2} are non-empty for each odd l ≥ 5.

(c) The elements in C̄_{2} are strongly real.

Proof : The first part follows from the previous theorem. As for part (b), the exceptional cases of Zsigmondy’s theorem do not occur here.

(c) We use the embedding of SO_{2l−1}(q) < SO_{2l}^{+}(q) to see that the elements we chose for C_{2} do lie in the dihedral subgroup for type B_{l−1} which we constructed in the proof of Theorem 4 resp. Theorem 3 if q is even.

2.5. The case D_{l}, l even

For G = SO_{2l}^{+}(q), l even, we cannot find a pair of maximal tori such that only two characters do not vanish on regular elements of both. But there exist such pairs with just three characters remaining. This will force us to make a particular choice for the third class to obtain the required result (although we feel that the result should remain true with C_{3} an arbitrary non-central class). Let T_{1}, T_{2} denote maximal tori of orders (q^{l−1} + 1)(q + 1), (q^{l/2} + (−1)^{l/2})^{2} respectively. Here gcd(|T_{1}|, |T_{2}|) = gcd(2, q − 1)^{2}. We define T′_{i} := T_{i} ∩ G′ with the normal subgroup G′ := Ω_{2l}^{+}(q) of index 2. Let C_{1}, C_{2} contain regular elements from the tori T′_{1} and T′_{2}. If q is odd then let C_{3} be the class of involutions with centralizer of type D_{l/2}(q)≀2 if q^{l/2} ≡ 1 (mod 4) or 2D_{l/2}(q) ≀ 2 if q^{l/2} ≡ −1 (mod 4) (see [25, Table 3.5.E]). If q is even, choose C_{3} to contain unipotent elements of type 2_{2}^{l} in the notation of [4, p.182]. These involutions lie already in G′ and have centralizer dimension l^{2} − l in the algebraic group [2, §8].

Theorem 2.7: (a) The normalized structure constant n(C̄_{1}, C̄_{2}, C̄_{3}) in Ḡ := O_{2l}^{+}(q), l even, is non-zero. (b) The classes C̄_{1} and C̄_{2} are non-empty for each even l with (l, q) ≠ (4, 2).

(c) The elements in C̄_{1} are strongly real.

Proof : (a) As above, we first determine the characters not vanishing on the first two classes, which again must be among the unipotent characters. The unipotent characters for groups of type D_{l} are parametrized by symbols of rank l and defect congruent 0 (mod 4) [4, 13.8]. Using [33] only the trivial character, the Steinberg character and the character corresponding to the symbol

Λ = \binom{0 … l/2 − 2 l/2 − 1 l/2 l − 1}{1 … l/2 − 2} for l/2 even,

Λ = ( \begin{matrix}0 & … & l/2 − 2 & l − 1 \\ 1 & … & l/2 − 1 & l/2 \end{matrix} ) for l/2 odd

are seen to be of interest. We obtain the following table of character values,

[1]  C_{1}  C_{2}  C_{3} (q odd)  C_{3} (q even)
1  1  1  1  1  1
ρ_{Λ}  ρ_{Λ}(1)  ±1  ±2  a_{1}  a_{2}
St  q^{l(l−1)}  1  1  q^{l(l−2)/2}  0

where the a_{i} remain to be determined. Clearly the structure constant is positive if |a_{i}| < ρ_{Λ}(1)/2. To prove this we will use the estimate |a_{i}|^{2} ≤ |G|/|C_{3}|, which is a consequence of the orthogonality relations.

First assume l/2 to be even and q to be odd. A formula for the degree of ρ_{Λ} is given in [4, 13.8]. Evaluating this and using the estimate (q^{n} − 1)/(q^{m} − 1) ≥ q^{n−m} for q > 1, n > m, we obtain

ρ_{Λ}(1) ≥ \frac{1}{2}q^{(3l^{2}−8l)/4}(q^{l/2} − 1)(q^{l/2−1} − 1)^{2}.

For the order of the centralizer C of elements in C_{3}, using the estimate q^{n} − 1 ≤ q^{n}, we see

|C| ≤ 8q^{l(l−1)}.

It follows that for l = 4 the required estimate holds if q ≥ 3, and for l ≥ 8 it holds for all q.

For l/2 odd, similar calculations yield the result. Also, in even characteristic the same type of estimates work for (l, q) ≠ (4, 2), yielding the required inequality. So the structure constant for our chosen classes does not vanish.

(b) By Zsigmondy’s theorem regular elements exist in T′_{1} for (l, q) ≠ (4, 2). The normalizer of T_{2} is isomorphic to a subgroup of index two in a wreath product of (q^{l/2} + 1).l with Z_{2}, where the l-cycle acts fixed point freely on q^{l/2} +1, as can be seen by embedding (q^{l/2} +1).l into B_{l/2}(q). Hence T_{2} contains regular elements if the l-cycle has at least two orbits on q^{l/2} + 1. For this we need l ≤ q^{l/2}/2, and this holds for all relevant (l, q).

(c) The same argument as in the proof of Theorem 2.6 applies.

2.6. The cases E_{6} and ^{2}E_{6}

Let G = E_{6}(q), q = p^{n}, be the simple group of type E_{6} defined over 𝔽_{q}. The conjugacy classes of G were determined by Mizuno [31]. Now we choose C_{1} to be a conjugacy class of G containing generators of the maximal torus T_{1} of type E_{6}(a_{1}) and of order Φ_{9}(q)/d = (q^{6} + q^{3} + 1)/d where d = gcd(q − 1, 3), C_{2} to contain elements of order Φ_{12}(q) = q^{4} −q^{2} +1 (so these are the generators of the Coxeter torus of a subgroup F_{4}(q)). By definition such elements exist for all q. Their centralizers coincide with the unique maximal tori containing them. The definition of the third class depends on whether involutions are semisimple or unipotent, i.e., whether q is odd or even.

In the first case G is known to possess two classes of involutions [31], with centralizers of type A_{5} + A_{1} and D_{5}. Let C_{3}^{−} be the second one. If q is even, by Mizuno’s work there exist three classes of involutions in G with representatives x_{1}, x_{3} and x_{6} [31]. Here we define C_{3}^{+} := [x_{6}]. Elements from this class have centralizer order q^{31}Φ_{1}^{3}Φ_{2}^{2}Φ_{3}/d.

Theorem 2.8: (a) The normalized structure constant n(C_{1}, C_{2}, C_{3}) in G = E_{6}(q) is non-zero. (b) The elements in C_{2} are strongly real.

Proof : (a) By work of Deligne and Lusztig it is known that only unipotent characters and characters R_{T_{1},θ} with θ in general position do not vanish on the class of the regular elements in C_{1}. So only these have to be considered in the evaluation of n(C_{1}, C_{2}, C_{3}). But C_{2} contains regular elements not lying in T_{1}, so the R_{T_{1},θ} vanish on C_{2} [4, 7.2.8]. A short inspection of the list of degrees of the unipotent characters [4, 13.9] shows that apart from 1, St, E_{6}[θ] and E_{6}[θ^{2}], they are of l-defect zero for a Zsigmondy prime l dividing either Φ_{9} or Φ_{12}, hence vanish on either C_{1} or C_{2} by a well known theorem of Brauer.

First assume q odd. Then the cuspidal unipotent characters E_{6}[θ] and E_{6}[θ^{2}] are of 2-defect zero, so take value zero on C_{3}. The values of the trivial and the Steinberg character are well known, see for example [4, Chap. 6] and by an argument as in the proof of Theorem 1 we verify n(C_{1}, C_{2}, C_{3}^{−}) > 0.

Now let q be even. Then St has 2-defect zero, and we are left with 1, E_{6}[θ] and E_{6}[θ^{2}]. Unfortunately the character theory for groups of type E_{6} in bad characteristic is not yet satisfactorily known to explicitly calculate the values of the two cuspidal characters on the unipotent class C_{3}. We denote the unknown entries by x_{1}, x_{2} in the following table. The values of the cuspidal unipotent characters on the regular semisimple elements in C_{1} and C_{2} are immediate from the character formula [4, 7.5.3].

[1]  C_{1}  C_{2}  C_{3}
|G|  Φ_{9}/d  Φ_{3}Φ_{12}/d  q^{31}Φ_{1}^{3}Φ_{2}^{2}Φ_{3}/d
ϕ_{1,0}  1  1  1  1
E_{6}[θ]  \frac{1}{3}q^{7}Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}  −1  1  x_{1}
E_{6}[θ^{2}]  \frac{1}{3}q^{7}Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}  −1  1  x_{2}

From elementary character theory we know that |χ(σ)|^{2} ≤ |C_{G}(σ)| for all σ ∈ G, so x_{1/2}^{2} ≤ q^{31}Φ_{1}^{3}Φ_{2}^{2}Φ_{3}/d, hence

|\frac{x_{1/2}}{E_{6}[θ](1)}|^{2} ≤ \frac{9q^{31}Φ_{1}^{3}Φ_{2}^{2}Φ_{3}}{dq^{14}Φ_{1}^{12}Φ_{2}^{8}Φ_{4}^{4}Φ_{5}^{2}Φ_{8}^{2}} ≤ \frac{9}{d} \frac{(q + 1)^{19}}{(q − 1)^{39}}

= \frac{9}{d} \frac{1}{(q − 1)^{20}} (1 + \frac{2}{q − 1})^{19} .

For q ≥ 4 this yields

|\frac{x_{1/2}}{E_{6}[θ](1)}|^{2} ≤ \frac{3}{d} (\frac{5}{9})^{19} < 1/3,

but it is immediately verified that this last inequality also holds for q = 2. With this estimate the first part of the Theorem follows.

(b) An element s of C_{2} is contained in the centralizer of the graph automorphism isomorphic to F_{4}(q) and generates the Coxeter torus of that group. As the Weyl group of F_{4} has a central involution, the Coxeter torus is contained in a dihedral group. So each element in C_{2} is strongly real.

It should be noted that the result carries over to the central cover E_{6}(q)_{sc}.

We now turn to the twisted case G = ^{2}E_{6}(q). By the general Ennola philosophy that ^{2}E_{6}(q) is E_{6}(−q), the result and the proof for the twisted case should carry over immediately from the previous one, and indeed this is the case. We choose C_{1} to contain generating elements from the torus T_{1} of order Φ_{18}/d, d = gcd(3, q + 1), and C_{2} again a class of generating elements of the Coxeter torus of a subgroup F_{4}(q), of order Φ_{12}. Again we have to distinguish between q odd and q even. In the first case we choose C_{3}^{−} to be the class of involutions with centralizer type ^{2}D_{5} [13]. For q even, let C_{3}^{+} be the class of involutions [v] described in the paper [2, 14.1]. So |C_{G}(v)| = q^{31}Φ_{1}^{2}Φ_{2}^{3}Φ_{6}/d.

Theorem 2.9: (a) The normalized structure constant n(C_{1}, C_{2}, C_{3}) in G = ^{2}E_{6}(q) is non-zero. (b) The elements in C_{2} are strongly real.

Proof : By the same arguments as before, we just have to consider the four unipotent characters 1, St, ^{2}E_{6}[θ] and ^{2}E_{6}[θ^{2}] for contributions to the structure constant. For odd q, the cuspidal characters have 2-defect zero and so vanish on C_{3}^{−}. For q even, the Steinberg character vanishes on the involutions in C_{3}^{+}. Estimates as in the previous proof now show n(C_{1}, C_{2}, C_{3}) > 0. The strong reality of elements in C_{2} again follows from the embedding of F_{4}(q), the centralizer of a graph automorphism.

3 Conclusion

Putting together the results of the previous two sections, we may state:

Theorem 3.1: Let G be a finite simple group of classical Lie type different from U_{3}(3), or an exceptional group of type E_{6} or ^{2}E_{6}. Then G may be generated by three of its involutions.

Proof : This follows by collecting the above results. Clearly all simple classical groups apart from U_{3}(3), O_{8}^{−}(2) and O_{8}^{+}(2) are covered by Theorems 2.1–2.7, since the exception (l, q) ≠ (2, 2) for S_{2l}(q) corresponds to the composite group S_{4}(2) ≅ S_{6}, and O_{2l+1}(2^{n}) is isomorphic to S_{2l}(2^{n}). We will now show that the respective triples of elements, for which the non-vanishing of the structure constant was proved, do generate the whole group (up to some exceptions which we treat separately). For type A_{1}(q), q > 7, q ≠ 9 (these restrictions may be made without loss of generality since A_{1}(4) ≃ A_{1}(5) ≃ A_{5}, A_{1}(7) ≃ A_{2}(2) and A_{1}(9) ≃ A_{6}), the generation follows from [14]. For the remaining groups we may apply Theorem 1.1 since in all cases one of the two regular semisimple elements was chosen to coincide with the particular element s̃ defined in Table 1.1. So we just have to verify that the maximal subgroups listed in Theorem 1.1 do not contain the second semisimple element.

First let G = A_{l}(q) with l ≥ 2 even. By Zsigmondy’s theorem and [25, 3.5.A] a primitive prime divisor of the order of an element in C_{2} does not divide the order of members from the collection C_{3} except possibly if l = 2 or q = 2, l = 6 (remember l ≥ 2). But in the latter cases we would have M = GL_{1}(q^{3}) or M = GL_{1}(2^{7}), and clearly these groups do not contain our second semisimple element. From cases (b) and (c) of Theorem 1.1 only (A_{2}(2), A_{2}(4)) has to be treated, and again the order of elements from C_{2} rules out this possibility. For A_{l}, l ≥ 3 odd, we just have to consider C_{1} (note that A_{3}(2) ≃ A_{8} does not need to be considered; in fact, one can check that A_{8} has a (2, 5, 7) generating system). But members of C_{1} are ruled out by a primitive prime divisor of elements from C_{1}.

Next we consider type ^{2}A_{l}, l even. By Zsigmondy the members of C_{3} may be excluded unless l = 2 or q = 2, l = 6. But then the structure of members of C_{3} yields a contradiction. This deals with case (a) of Theorem 1.1. From (b) we have to consider here (A_{7},^{2}A_{2}(5)) and (A_{1}(11),^{2}A_{4}(2)). In the second case we may choose our involution to be a transvection, and since A_{1}(11) is not a transvection group, we get generation. In the first case, we take the class triple (2A, 5B, 7A) instead of the one in Theorem 2.2; here the second class is strongly real. Then from [6] it is possible to check that there exist generating triples in this class vector. This completes ^{2}A_{l}, l even. If l is odd, the only Zsigmondy exception occurs for q = 2, l = 5. Here we choose the class triple (11A, 15A, 2C). Then its class number is different from zero [6], and the involutions from 2C do not lie in the maximal subgroup U_{5}(2). The only other maximal subgroup with order divisible by 11 is M_{22}, but that does not contain elements of order fifteen. Finally, elements from 15A are strongly real, proving the assertion for ^{2}A_{5}(2). From parts (b) and (c), we have to consider (A_{7},^{2}A_{3}(3)), (A_{2}(4),^{2}A_{3}(3)) and G ≃ ^{2}A_{3}(2). For ^{2}A_{3}(3) we choose (7A, 5A, 2A), and then the result follows with [6]. The group ^{2}A_{3}(2) is isomorphic to S_{4}(3) and will be treated there.

Now let G be of type B_{l}(q), l ≥ 3, q odd. The only possibility from class C_{1} in [1] is O_{1}(q) × O_{2l}^{−}(q). But the order of this group is not divisible by a primitive prime divisor of q^{l} − 1, so case (a) of Theorem 1.1 does not apply. The only other possibility is (S_{9}, B_{3}(3)), but here our second element has order 13, which does not divide |S_{9}|. This completes the proof for type B_{l}.

If G has type C_{l}(q), l ≥ 2, several cases have to be considered. For q even, the generation follows from [35, Satz 2.21]. So let now q be odd. Then members of C_{3}, isomorphic to S_{2l/k}(q^{k}).k, k|2l prime, have to be excluded. For that, we choose a suitable involution for our third class C_{3}. Namely, for each 1 ≤ m ≤ l there exist involutions in Sp_{2l}(q) with 2m eigenvalues −1. In subgroups Sp_{2l/k}(q^{k}).k we only need to consider involutions lying already inside Sp_{2l/k}(q^{k}), because otherwise the (C_{1}, C_{2}, C_{3}) structure constant for the subgroup vanishes anyway. Clearly the multiplicity of the eigenvalue −1 is divisible by r for all such involutions. We now choose m < l with m|\ l, which is possible for l ≠ 2, and take C_{3} to contain involutions such that their preimage in the symplectic group has 2m eigenvalues −1. If l = 2, S_{2}(q^{2}) = L_{2}(q^{2}) contains one class of involutions, while S_{4}(q) has two, so again we may choose C_{3} outside that subgroup. This eliminates the cases in C_{3}.

From (b) and (c) arises only C_{2}(3) = ^{2}A_{3}(2) (remember that q is odd). For this last group, the class vector (2C, 9A, 10A) contains generating triples, and 9A is strongly real, as follows readily from [6].

For type ^{2}D_{l}, l ≥ 4, the exceptions in C_{3} are O_{2l/k}^{−}(q^{k}) for k|2l, k > 1, or GU_{l}(q).2 (see Table 1.3), but their order is not divisible by a primitive prime divisor of q^{l−1} + 1, except possibly if q = 2, l = 4. There we didn’t even yet find a suitable class triple. We choose (17A, 21A, 2A) in Atlas notation, then the structure constant does not vanish, the classes 17A and 21A are both strongly real, and from the list of maximal subgroups in [6] it follows that we also have generation.

If G has type D_{l}(q) for l odd, primitive prime divisors of q^{l} − 1 exclude the groups in C_{1}, C_{3}, and these are the only ones to consider by Theorem 1.1.

For type D_{l}(q), l even, we notice that the groups in class C_{1} act reducibly on the natural module, while any group generated by regular elements in T_{1} and T_{2} clearly acts irreducibly. Of the groups in C_{3} only GU_{l}(q) remains by Zsigmondy’s theorem applied to T_{1}. But the regular element in T_{2} has centralizer order (q^{l/2} + 1)^{2} in G. If it were contained in GU_{l}(q), a subgroup of G of the same rank embedded in the natural way, then its centralizer order in GU_{l}(q) would have to coincide with the one in G. This is a contradiction as follows by comparing the multiplicity of a primitive prime divisor of q^{l/2} + 1 in both centralizers. It hence remains to treat the exceptions in Theorem 1.1(b)(ii). By [23] for each embedding as in (b)(ii) there exists an 8-dimensional representation of G such that the cited subgroups act reducibly, again contradicting the fact that the group generated by our first two elements has to act irreducibly. So we are left with the single group D_{4}(2), for which even Theorem 2.7 failed. There the class triple (2B, 9A, 15B) has a non-zero structure constant, and with the permutation characters in [6] it is possible to check that such triples are contained in no proper maximal subgroup. Moreover elements in 9A and 15B are strongly real. This completes our case by case analysis for the classical groups.

For G = E_{6}(q) the only proper subgroups containing the torus T generated by an element from C_{1} are already contained in subgroups of type P SL_{3}(q^{3}).3 [36], and so by Zsigmondy their order is not divisible by Φ_{12}. This proves generation for E_{6}(q). Similarly the result follows for ^{2}E_{6}(q). (The cases q = 2, 3 are not covered by [36] but may easily be treated as in [28], proof of Theorem 3.3.)

In all cases, C_{3} may be (or was already) chosen to contain involutions. If C denotes the strongly real class among C_{1} and C_{2}, then clearly also a pair of elements from (C, C_{3}) generate G. Replacing the element from C by a product of two involutions we obtain the required assertion about generation by three involutions.

Together with the results of [17, 7, 36] for the sporadic groups, the alternating groups and the exceptional groups of Lie type different from E_{6} and ^{2}E_{6} we conclude:

Corollary: Every finite non-abelian simple group apart from U_{3}(3) is generated by three involutions.

Proof of Theorem B: The assertion of Theorem B for all classical simple groups and for the exceptional groups of type E_{6} is contained in the proof of Theorem 3.1. For the remaining exceptional groups, it follows from the main result of [36].

For G ≃ A_{n}, n ≥ 5, every cycle of odd length k lies in a dihedral group of order 2k contained in S_{n}. This implies that every element g ∈ A_{n} such that ord(g) ≡ 1 (mod 4) is a strongly real element in A_{n}. A theorem of G. A. Miller [30] states that for every number s > 3 for which A_{n} contains elements of order s there exists an element g of that order and an involution h such that A_{n} = ⟨g, h⟩. This shows that A_{n}, n ≥ 5 is generated by an involution and a strongly real element.

Finally, for the sporadic groups there exist numerous results on particular types of generation, many of which imply our result. Here we use the list given in [5] of minimal generating systems. For all sporadic groups G apart from M_{24}, F i_{23}, B and M, this yields a (2, n_{1}, n_{2})-generating system (with product of the generators equal to 1), and from the Atlas [6] it is easily verified in all cases apart from F i′_{24} that one of the two further elements is strongly real. The group M_{24} is (2, 10, 23)-generated, with a strongly real element of order 10. F i_{23} is (2, 17, 23)-generated, F i′_{24} is (2, 17, 29)-generated, B is (2, 17, 47)-generated, all with a strongly real 17-element, and finally M has a (2, 41, 71)-generation, with strongly real element of order 41, as can be checked from the Atlas.

The results of the previous sections also allow a quite different application. Thompson asked whether each finite simple group contains a conjugacy class C such that the set of products C^{2} of elements of C covers G. From our results we can prove at least the weaker statement for the cube of a class:

Theorem 3.2: Let G be a finite simple group of classical Lie type different from D_{2l}(q). Then there exists a conjugacy class C of G such that C^{2} ∪ C^{3} covers G.

Proof : In Theorems 2.1–2.6, the third class C_{3} could be chosen arbitrarily, distinct from the trivial class. One of C_{1}, C_{2} always contained strongly real elements; denote this class by C, and the other one by C′. As all classes (except the trivial one) intersect C · C′, we see that elements from C′ may be written as the product of two elements from C. Hence all non-identity classes lie in C^{3} (while the trivial class already lies in C^{2} since C contains the inverses of all its elements).

For U_{3}(3), we may choose C = 3B in Atlas notation, and then even C^{2} covers the whole group. For O_{8}^{−}(2), the square of class 17A does the job by [6].

It is not clear how to extend this last application to D_{2l}(q) with our methods; one way might be to prove that Theorem 2.7 holds without restriction on C_{3}.

Remark: After completing this paper, the authors learned of an announcement of Nuzhin, where he claims to prove a result similar to Theorem 3.1 for groups of Lie-rank at least four.

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