Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 349-358.

11 ; − ; 2-Generation of Finite Simple Groups

Alexander Stein

Institut für Algebra und Geometrie, FB Mathematik und Informatik Martin-Luther-Universität Halle-Wittenberg Theodor-Lieser-Straße 5, D-06099 Halle, Germany

In 1962 Robert Steinberg showed in [11], that each finite simple group of Lie-type is generated by two elements, using his generators and relations for this groups. He asked whether or not each finite simple group G can be generated by two elements, one an arbitrary nonidentity element of G. For alternating groups this was already known. Using the classification of finite simple groups we can now state:

Theorem. Every finite simple group G can be generated by two elements, one an arbitrary nonidentity element, the other one from a fixed conjugacy class only depending on G.

The theorem holds for large classes of finite simple groups by work of other authors such as alternating groups (Piccard [10](1946)), (Brenner and Wiegold [1](1975), [2](1980)), sporadic groups (Woldar [13](1995)) and some classical groups (Malle, Saxl and Weigel [8](1994)).

After finishing the proof we came across with a preprint due to R. Guralnick and W. Kantor [4]. They proved a much stronger theorem, which uses some hard arguments concerning the number of fixed points in some permutation representations of finite simple groups. Both papers heavily depend on the work of G. Malle, J. Saxl and T. Weigel [8], [12]. But in contrast to [4] our proof just uses elementary arguments beside the use of [8] and [12].

As the theorem obviously is equivalent to the existence of a y ∈ G such that no whole nontrivial conjugacy class of G lies in the union of all maximal subgroups of G containing y, the argumentation is as follows:

For an element y ∈ G all maximal subgroups of G containing y are determined. (In fact this was already done in [12] and [8].) Now one has to prove, that no whole conjugacy class of G lies in the union of these maximal subgroups. Now we are done.

The author gratefully acknowledges the support of Prof. G. Stroth.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

Alexander Stein: 1\frac{1}{2}-Generation of Finite Simple Groups 350

Remark. If a conjugacy class C of G lies in a union of subgroups, there exists a conjugacy class D, which also lies in the union of these subgroups and elements of D have prime order.

Proposition 1. Let G be a finite group and C any nontrivial conjugacy class with < C >= G. Then C cannot lie in the union of two proper subgroups.

Proof. Assume false and let U, V be the subgroups with C ⊂ U ∪ V. Let C_{V} = C − U ⊂ V and C_{U} = C − V ⊂ U. The sets C_{U} and C_{V} are nonempty for otherwise C is contained in one of the proper subgroups U or V but < C >= G.

C_{U} is normalized by V because C_{U} is the set of all elements of C outside of V. Thus H :=< C_{U} > V is a subgroup of G. Because C_{V} is nonempty and < C_{U} > ◁H, < C_{U} >≤ U there exist elements x ∈ C_{U}, y ∈ C_{V}, which are conjugate in G, but not in H. Thus H is a proper subgroup of G, contradicting < C >= G, as C ⊂ H. □

Proposition 2. Let G be a finite group, P_{1}, P_{2}, N_{G}(L) maximal subgroups of G which satisfy the following conditions: L := P_{1} ∩ P_{2}, |N_{G}(L) : L| = 2, P_{1}, P_{2} are the only conjugates of P_{1} containing L. If core_{G}(L) = 1 and a conjugacy class C of elements of prime order with < C >= G lies in the union of this three maximal subgroups, then G is a symmetric group and C is a class of transpositions.

Proof. If the elements of C have odd order and C lies in P_{1} ∪ P_{2} ∪ N_{G}(L), then C already lies in P_{1} ∪ P_{2} as |N_{G}(L) : L| = 2 and L = P_{1} ∩ P_{2}. By Proposition 1 this is impossible. So C is a class of involutions. Let Ω be the set of all conjugates of P_{1} and identify this set with natural numbers, s.t. P_{1} = 1 and P_{2} = 2. Because of P_{1} = N_{G}(P_{1}) (P_{1} maximal) and 1 = core_{G}(L) = core_{G}(P_{1}) the operation of G on Ω is transitive and faithful. By assumption every element x of C has one of the following types: (α): x = (1a)(2) …, a ≠ 1, 2, (β): x = (2b)(1) …, b ≠ 1, 2, (γ): x = (12) …, (ϵ): x = (1)(2) … Because of Proposition 1 there exist elements of the types (α), (β), (γ).

Let i be an element of type (γ) and B_{1}^{2}, B_{2}^{1} be the orbits of Stab_{1} = P_{1} resp. Stab_{2} = P_{2}, containing 2 resp. 1. Assume i moves a point a ∈ B_{1}^{2} −{2}. By definition there is a g ∈ Stab_{1} with g(1) = 1 and g(a) = 2, so i^{g} has type (1g(2))(2g(i(a))) … and has none of the above types, a contradiction. So i is trivial on B_{1}^{2} − {2} and symmetrically trivial on B_{2}^{1} − {1}.

Let a ∈ B_{1}^{2} − {2}. Then there is an element h with h(1) = 1 and h(2) = a. Now h^{i}(2) = 2 and h(1) = a, so B_{1}^{2} ∪ {1} = B_{2}^{1} ∪ {2} =: Ω′ is invariant under P_{1}, P_{2}, i so under < P_{1}, P_{2}, i >= G but G is transitive. So Ω′ = Ω and i is a transposition. Using transitivity and generation by transpositions one gets G ≅ Σ_{n} for n = |G : P_{1}|. □

Proposition 3. Let y ∈ G and M_{i}, i ∈ I, be all maximal subgroups of G, containing y. For J ⊂ I define D_{J} = ⋂_{i∈J} M_{i} and assume D_{J} < D_{K} for every proper subset K of J. Let C be a conjugacy class with C ⊂ ⋃_{i∈I} M_{i} and < C >= G. Then there is a x ∈ C with G =< x, y >.

Proof. Assume C lies in the union of the maximal subgroups M_{i}, i ∈ I. Choose x_{1} in C such that l = |L| = |{j ∈ I : x_{1} ∈ M_{j}}| is minimal. Assume l < |I|. Set K = I − L and C_{1} = C − ⋃_{k∈K} M_{k}. Every element x_{2} ∈ C_{1} lies in at least l subgroups M_{j}, but j ∉ K, so j ∈ L and C_{1} ⊂ D_{L}. Because of x_{1} ∈ C_{1}, C_{1} is nonempty. Let x_{3} ∈ C_{1} and v ∈ D_{K}. Because x_{3}^{v} ∈ C there exists a j ∈ I with x_{3}^{v} ∈ M_{j}. Assume j ∈ K. Because v ∈ D_{K} ≤ M_{j} and x_{3} ∉ M_{j} this is a contradiction. So j ∈ L and by minimality of l x_{3}^{v} is in C_{1}. So D_{K} permutes C_{1} and normalizes N :=< C_{1} >≤ D_{L} < G. Set H := ND_{K} and assume H = G. Then < C_{1} >= N ◁ H = G, which contradicts < C >= G.

Because y ∈ D_{K} < H, H is contained in M_{j} for some j ∈ I. Because x ∈ M_{j} it follows j ∈ L but then D_{K} ≰ M_{j} because D_{K} > D_{K∪{j}} = D_{K} ∩ M_{j}. This contradiction shows, that l = |I|, so C ⊂ D_{I}, which contradicts G =< C >. This final contradiction shows the existence of an x ∈ C, such that < x, y > never lies in a maximal subgroup of G containing y, so < x, y >= G. □

Alternating and sporadic groups

Lemma 1. Let G be alternating or sporadic. Then there is a y ∈ G, such that for any 1 ≠ x ∈ G a conjugate y^{g} exists with G =< x, y^{g} >.

Proof. For alternating groups the theorem is a classical result. An old proof is in [10], a newer one follows for instance from Theorem 3.05 in [1] and Theorem 2.3 in [2] for even/odd degree respectively. The sporadic groups were recently handled by Woldar in [13]. □

Exceptional groups

Lemma 2. Let G be an exceptional simple group of Lie-type. There exists a maximal abelian p′-subgroup T of order given in Table 1. T is either cyclic and then y is defined as a generator of T or G = E_{7}(q). Every maximal subgroup of G containing T is either listed in the third column or G is one of the following groups: ^{2}F_{4}(2)′, G_{2}(3), G_{2}(4), F_{4}(2). In this list τ means a graph-automorphism of G, which normalizes < y > (if it exists), P_{1} in E_{7}(q) means a parabolic subgroup of type E_{6}(q) and i is an involution of N_{G}(T) − T in case of E_{7}(q).

Proof. This is exactly the statement in [12] about the maximal subgroups of G containing T. The statement in [12] even holds for the case G = F_{4}(3), ^{2}E_{6}(2) and ^{2}E_{6}(3), which one can check using the classification (as only the nonlocal case has to be checked). □

Lemma 3. Let G be an exceptional simple group of Lie-type. There exists a conjugacy class C, such that for any 1 ≠ x ∈ G there is a y ∈ C with < x, y >= G.

Proof. Let 1 ≠ x ∈ G. First let G be none of the exceptional cases of Lemma 2 and G ≠ E_{7}(q). By Lemma 2 there are at most two maximal subgroups containing y, so by Proposition 1 and simplicity of G there is a conjugate x^{g} of x with < y, x^{g} >= G, so G =< y^{g^{−1}}, x >. Let G = E_{7}(q). If q = 3^{n} let y be an element of order (q − 1)/(q − 1, 2)(q^{6} + q^{3} + 1) for n > 1 and of order 2(3^{6} − 3^{3} + 1) if q = 3. If (q − 1, 3) = 3 let y be an element of order (q + 1)/(q − 1, 2)(q^{6} − q^{3} + 1) and if (q + 1, 3) = 3 let y be an element of order (q − 1)/(q − 1, 2)(q^{6} + q^{3} + 1). By [12] such a y always exists.

G  |T |  Maximal subgroups

containing T

2B_{2}(2^{2n+1}) 2^{2n+1} + 2^{n+1} + 1  N_{G}(T)
2G_{2}(3^{2n+1}) 3^{2n+1} + 3^{n+1} + 1  N_{G}(T)
2F_{4}(2^{2n+1}) 2^{4n+2} + 2^{3n+2} + 2^{2n+1} + 2^{n+1} + 1 N_{G}(T)
G_{2}(q)  q^{2} − q + 1  M ≅ SU(3, q).2, (M^{τ})
3D_{4}(q)  q^{4} − q^{2} + 1  N_{G}(T)
F_{4}(q)  q^{4} − q^{2} + 1  M ≅ ^{3}D_{4}(q).3, (M^{τ})
E_{6}(q)  \frac{1}{(3,q−1)}(q^{6} + q^{3} + 1)  P SL(3, q^{3}).3
2E_{6}(q)  \frac{1}{(q+1,3)}(q^{6} − q^{3} + 1)  P SU(3, q^{3}).3
E_{7}(q)  \frac{1}{(q−1,2)}(q − 1)(q^{6} + q^{3} + 1)  P_{1}, P_{1}^{i}, N_{G}(P_{1} ∩ P_{1}^{i})
\frac{1}{(q−1,2)}(q + 1)(q^{6} − q^{3} + 1)  (\frac{q+1}{(q−1,2)}^{2}E_{6}(q)).2
E_{8}(q)  q^{8} + q^{7} + q^{5} − q^{4} − q^{3} + q + 1  N_{G}(T)

Table 1: Maximal subgroups of exceptional groups of Lie-type containing T

If y has order (q + 1)/(q − 1, 2)(q^{6} − q^{3} + 1) there is by Lemma 2 just one maximal subgroup of G containing y, in the other case the maximal subgroups of G containing y satisfy the conditions of Proposition 2, but G ≠ Σ_{n}. So in both cases there exists a conjugate x^{g} ∈ G with G =< y, x^{g} > and G =< x, y^{g^{−1}} >. So let G be one of ^{2}F_{4}(2)′, G_{2}(3), G_{2}(4), F_{4}(2). The first two cases one can check by computer: For a small permutation- or matrixrepresentation of G a set of representatives for the conjugacy classes of elements of prime order of G is calculated. (In case of G_{2}(3) was the 7-dimensional representation over GF(3) used given by the Atlas of Modular Representations (http://for.mat.bham.ac.uk/atlas/), in case of ^{2}F_{4}(2)′ a permutation representation on 1755 points given in the computer algebra system MAGMA.) The representatives can be chosen, such that any representative and a fixed y ∈ G, here both of order 13, generate G, which one checks also by computer.

In case of G = G_{2}(4) there are by [3] exactly two subgroups of G containing an element of order 13 and by Proposition 1 the statement holds.

So G = F_{4}(2). Let y be an element of order 17. By [9] there are two conjugacy classes of maximal subgroups divisible by 17, both isomorphic to Sp_{8}(2), so N_{G}(< y >) is contained in any maximal subgroup containing y, so y is contained in exactly 2 maximal subgroups. By Proposition 1 now there is an x^{g} ∈ G with G =< x^{g}, y >=< x, y^{g^{−1}} >. □

Classical groups

Definition. In the following let G be a finite simple classical group of Lie-type and y an element of the order given in Table 2. If G = P Ω^{+}(2n, q) and n ≥ 6 let y be an element of the given order, which acts fixed point freely on its natural projective module. So y is chosen as in [8].

Lemma 4. Let M be a maximal subgroup of G containing y. Then either M is contained in Table 3 or G is one of the following groups: P SL(2, q), P SL(3, 4), P SL(4, 2), P SU(3, 3),

G  Conditions o(y)
P SL(n + 1, q) n even  (q^{n+1} − 1)/((q − 1)(q − 1, n + 1))
n ≥ 3 odd (q^{n} − 1)/(q − 1, n + 1)
P SU(n + 1, q) n even,  (q^{n+1} + 1)/((q + 1)(q + 1, n + 1))
n ≥ 3 odd (q^{n} + 1)/(q + 1, n + 1)
P Ω(2n + 1, q) n ≥ 3, q odd (q^{n} + 1)/2
P Sp(2n, q)  n ≥ 2  (q^{n} + 1)/(2, q − 1)
P Ω^{+}(2n, q)  n ≥ 4  (q^{n−1} + 1)/(2, q − 1)
P Ω^{−}(2n, q)  n ≥ 4  (q^{n} + 1)/(4, q^{n} + 1)

Table 2: The order of y in classical simple groups

P SU(3, 5), P SU(4, 2) ≅ P Sp(4, 3), P SU(4, 3), P SU(5, 2), P SU(6, 2), P Sp(8, 2), P Ω(7, 3), P Ω^{+}(8, 2). P_{i} means a stabilizer of a subspace of dimension i, N_{i}^{ϵ} means a stabilizer of a nondegenerate subspace W of dimension i, with W^{⊥} of ϵ-type in the odd-dimensional orthogonal groups.

Proof. This is Theorem 1.1 in [8]. □

G  type of maximal subgroup conditions
P SL(n + 1, q) GL(\frac{n+1}{k}, q^{k}).k  k prime, k|n + 1, n even
P_{1}, P_{n}  n odd, n ≥ 3
P SU(n + 1, q) GU(\frac{n+1}{k}, q^{k}).k  k prime,k|n + 1, n even
N_{1}  n ≥ 3 odd
P Ω(2n + 1, q) N_{1}^{−}  n ≥ 3, q odd
P Sp(2n, q) Sp(\frac{2n}{k}, q^{k}).k, GU(n, q).2 qn odd,k prime, k|n,n > 2
Sp(\frac{2n}{k}, q^{k}).k  q odd,k prime, k|n,n ≥ 2 even
Sp(\frac{2n}{k}, q^{k}).k, O^{−}(2n, q)  q even,k prime, k|n,n ≥ 2
P Ω^{+}(2n, q) N_{2}^{−}, GU(n, q).2 n ≥ 6, even
N_{2}^{−}, O_{n}(q^{2}).2  n ≥ 5 odd
(N_{2}^{−})^{τ^{k}}, N_{1}^{τ^{k}}  n = 4, τ some triality
P Ω^{−}(2n, q) O^{−}(\frac{2n}{k}, q^{k}), GU(n, q).2, k prime, k|n, n ≥ 4 even,
O^{−}(\frac{2n}{k}, q^{k})  k prime, k|n, n > 4 odd

Table 3: The maximal subgroups of classical simple groups containing y

Lemma 5. Let G be a finite simple group of type P SL(n, q), P SU(n, q), P Ω(2n + 1, q), P Ω^{−}(2n, q) or P Ω^{+}(4m + 2, q) and y as defined or y of order (q + 1)/(q − 1, 2) in case P SL(2, q). Then for any 1 ≠ x ∈ G there is a conjugate y^{g}, such that G =< x, y^{g} >.

Proof. First let G be one of U_{5}(2), P Ω(7, 3), P SL(3, 4) or any classical simple group not in the list of exceptions of Lemma 4. By [8], theorems 2.1, 2.2, 2.4, 2.5 and 2.6 there is a z ∈ G, such that G = {y^{g}z^{h}|g, h ∈ G} ∪ {1}. So x = y^{g}z^{h} for some g, h ∈ G. The order of z is given in Table 4. So one can show, using Lemma 4 or [3] in case of the exceptions of Lemma 4, that G =< y, z^{h} > for any h ∈ G. Because z^{h} ∈< x, y^{g} > and < y^{g}, z^{h} >= G also < x, y^{g} >= G. So let G one of the exceptions of Lemma 4 not handled above. If G is of type P SL(2, q), the maximal subgroups of G containing y are very well known. In case q ≠ 4, 7, 9 the only maximal subgroup containing y is a dihedral subgroup of order (2, q)(q + 1). In case q = 7, 9 there are at most two maximal subgroups of G containing y, so by Proposition 1 we are done and in case q = 5 the conditions of Proposition 2 are satisfied, but G is not symmetric.

In case P SU(3, 3) there is by [3] exactly one maximal subgroup containing y, so we are done. The cases P SU(3, 5), P SU(4, 2), P SU(4, 3) and P SU(6, 2) could be verified using a computer, as in the case of the exceptional groups of Lie-type. (The representations used were the permutation representations on the isotropic points, given by the computer algebra system MAGMA.) The case of P SL(4, 2) = A_{8} was already handled in Lemma 1. □

G  Conditions o(z)
P SL(n + 1, q) n even  (q^{n/2} + 1)/(2, q − 1)
n ≥ 3 odd (q^{(n+1)/2} + 1)
P SU(n + 1, q) n even  (q^{n/2} + (−1)^{n/2})
n ≥ 3 odd, (q^{(n+1)/2} + (−1)^{(n+1)/2})/(2, q − 1)
P Ω(2n + 1, q) n ≥ 3, q odd (q^{n} − 1)/2
P Ω^{+}(2n, q)  n > 4 odd (q^{n} − 1)/(q^{n} − 1, 4)
P Ω^{−}(2n, q)  n ≥ 4  (q^{n−1} + 1)(q − 1)/(q − 1, 2)

Table 4: The order of z in classical simple groups

Lemma 6. Let G = P Sp(2n, q), n ≥ 2 and G simple. Then for each 1 ≠ x ∈ G there exists a y^{g} of order given in Table 2 with G =< x, y^{g} >.

Proof. By Lemma 4 the maximal subgroups of G containing y are given in Table 3 or G = P Sp(8, 2) or G = P Sp(4, 3). The case of P Sp(4, 3) was checked on a computer together with the case U_{4}(2) (note that the elements y given in Table 2 are different), in case P Sp(8, 2) [3] gives the maximal subgroups containing y of order 17. These are of type O^{−}(8, 2), P Sp(4, 4).4 and P SL(2, 17).

In all cases from each of these conjugacy classes of maximal subgroups there is exactly one maximal subgroup containing y, because there is a Zsygmondy-prime r dividing o(y) and each maximal subgroup containing y contains N_{G}(R) for some R ∈ Syl_{r}(G). We show now, that all intersections of the maximal subgroups containing y are known and satisfy the conditions of Proposition 3.

Let k = GF(q) and k ≤ K = GF(q^{n}). For any subfield L between k and K define subgroups of type A(L) := Sp(2|K : L|, |L|).Gal(L/k), B(L) := O^{−}(2|K : L|, |L|).Gal(L/k) in case q even and B(L) := U(|K : L|, |L|).2Gal(L/k) in case qn odd. Let Π denote the set of all primes dividing n, and n_{0} their product. For m dividing n denote with k[m] the (unique) subfield of K of degree m over k. So in general the maximal subgroups of G containing y are of type A(k[r]) and B(k), where r ∈ Π. For a type T denote with M_{T} a subgroup of G containing y of type T. Then the following holds for ∅ ≠ Ψ ⊆ Π:

(i) ⋂_{r∈Ψ} M_{A(k[r])} = M_{A(k[n_{0}|_{Ψ}])} (ii) M_{B(k)} ∩ ⋂_{r∈Ψ} M_{A(k[r])} = M_{B(k[n_{0}|_{Ψ}])} (iii) The conditions of Proposition 3 are satisfied or G = P Sp(8, 2).

Proof of the claim: Denote with D the intersection of all maximal subgroups of type A or B containing y. For (i) and (ii) we have only the inclusion ” ⊇ ” to show, as we know by [6] that the intersection of any two overgroups of D is maximal in both of them if the inclusion ” ⊇ ” in (i) and (ii) holds.

For fields E ≤ F denote with T_{E}^{F} the tracemap from F to E. Let f be a nondegenerate symplectic form on the F -space V. Then T_{E}^{F} f is a nondegenerate symplectic form on the natural E-space V. This shows, that there exist subgroups of type A(k[r]) in G and for any ∅ ≠ Ψ ⊆ Π there exist an embedding of groups of type A(k[n_{0}|_{Ψ}]) into subgroups of type A(k[r]) for r ∈ Ψ. But all these different embeddings lead to the same embedding into G as the equation T_{E}^{F} T_{F}^{H} = T_{E}^{H} for fields E ≤ F ≤ H holds. So we got the inclusion ” ⊇ ” in (i).

Let q even and V be a 2-dimensional K-space with a nondegenerate orthogonal form of minus-type Q. Then f(u, v) = T_{k}^{K}(Q(u + v) + Q(u) + Q(v)) is a nondegenerate symplectic form on the k-space V. Also if E ≤ F are fields and Q is a nondegenerate orthogonal form of minus-type over the F -space V, then T_{E}^{F} f is a nondegenerate symplectic form of the E-space V linked to the nondegenerate orthogonal form of minus-type T_{E}^{F} Q. Arguing like in the proof of (i) we get (ii).

So let qn odd. Denote with K_{1} an (up to ismorphisms unique) field of degree 2 over K and k_{1} the unique subfield of K_{1} of degree 2 over k. Let λ ∈ k_{1} with λ̄ = −λ. For a nondegenerate unitary form f of an F_{1}-space V the map T_{F}^{F_{1}}(λf) is a nondegenerate symplectic form of the natural F -space V. Also for a unitary form f and a subfield E_{1} of F_{1} the map T_{E}^{F} f is a unitary form of an E_{1}-space. As |K_{1} : k_{1}| = |K : k| = n is odd we can show as in (i), that all probably different embeddings of subgroups of type A and B lead to the same embeddings, which shows (ii). Now by (i), (ii) and Lemma 4 holds (iii) except in case P Sp(4, 3) and P Sp(8, 2).

Now Proposition 3 gives us a conjugate x^{g} with G =< x^{g}, y >=< x, y^{g^{−1}} >.

In case Sp(8, 2) we have three maximal subgroups and by (ii) the maximal subgroups of type O^{−}(8, 2) and Sp(4, 4).2 have a subgroup O^{−}(4, 4).2 = L_{2}(16).4 in common. A small modification of Proposition 3 gives us then a conjugate x^{g} with G =< x^{g}, y >. (If a conjugacy class C lies in these three subgroups, the L_{2}(16).4 normalizes all elements of C outside of O^{−}(8, 2) and Sp(4, 4).2, but all these elements lie by assumption inside of the L_{2}(17). So we can construct a new maximal subgroup containing y, a contradiction.) □

Lemma 7. Let G = P Ω^{+}(4m, q) with m ≥ 2 and y as defined. Then for any 1 ≠ x ∈ G there exists a conjugate y^{g}, such that G =< x, y^{g} >.

Proof. First let n = 2m ≥ 6. By definition y is chosen to fix no nonsingular points. (There exists a subgroup of type O^{−}(2, q) ⊥ O^{−}(2n − 2, q) inside of O^{+}(2n, q). Take an element y_{1} of order q + 1 inside of the O^{−}(2, q) and y_{2} of order q^{n−1} + 1 inside of O^{−}(2n − 2, q). Then the image of y_{1}y_{2} in P Ω^{+}(2n, q) fixes no point of the projective module, even for q = 3 where y is a product of an involution acting fixed point freely on the projective points of a 2-space and an element of odd order acting irreducibly on the 2n − 2-dimensional orthogonal complement.) There are three subgroups containing y, one (M_{1}) of type N_{2}^{−} (stabilizer of a 2 ⊥ 2n − 2 decomposition of the natural projective module V) and two (M_{2} and M_{3}) of type GU_{n}(q), conjugate under a graph automorphism. This follows from Lemma 4 together with the observation that all these maximal subgroups contain N_{G}(R) for R ∈ Syl_{r}(G) for a Zsygmondy-prime r dividing o(y).

Assume a nontrivial conjugacy class C of G is contained in these three subgroups. Let U be the 2-dimensional subspace of V, stabilized by y (and M_{1}). Let N_{2}^{−} be the set of all nondegenerate 2-subspaces of V of minus-type. Then U^{C} ⊂ N_{2}^{−} is a union of orbits of M_{1} on N_{2}^{−}: For let W = U^{x} a 2-dimensional subspace with x ∈ C and g ∈ M_{1}. Then U^{g^{−1}xg} = U^{xg} = W^{g}.

Let f(u, v) be a unitary form on an l-dimensional vectorspace over GF(q^{2}). Then Q(u) := f(u, u) is an orthogonal form over the field GF(q). This shows the embedding of GU_{n}(q) into O_{2n}^{+}(q) with n even and the following: Let V̂ be the unitary space and V the orthogonal space. An isotropic point of V̂ is a total singular 2-subspace of V and a nonisotropic point of V̂ is a total nonsingular 2-subspace of V. Every orthogonal point lies in a unique unitary point and two different unitary points have no orthogonal point in common. By the action of M_{2} on V̂, we see that U is a nonisotropic point. Let C be the conjugacy class of G, contained in M_{1} ∪ M_{2} ∪ M_{3} and x ∈ C with W := U^{x} ≠ U. So x is contained in one of the unitary groups and W ∩ U = 0, because W and U are different unitary points. So the number |U^{C}| is bounded by twice the number of points of an n-dimensional (unitary) space over GF(q^{2}) which is less than s_{0} := 2(q^{2n} − 1)/(q^{2} − 1). We will show now that no such orbit of M_{1} exists: Let U_{1} = U^{⊥} ∩ (W + U) ≤ V̂. Then U_{1} is a unitary point, so either a 2-dimensional total nonsingular or total singular space. But U_{1} and U are orthogonal also in V. As M_{1} is transitive on both the total singular and total nonsingular 2-subspaces of U^{⊥}, the orbit of M_{1} containing U_{1} has size at least

s_{1} := min ((q^{n−1} + 1)(q^{n−3} − 1)(q^{2n−4} − 1)/((q − 1)(q^{2} − 1)) , 1/2q^{2n−4}(q^{n−1} + 1))

The orbit of M_{1} on W has size at least s_{1}, because the map z ∈ M_{1} mapping U_{1} to U_{1}^{z} ≠ U_{1} maps W to W^{z} ≠ W, as U ⊥ U_{1} = U +W. By comparing s_{1} and s_{0} we get the contradiction, that in praxis s_{1} > s_{0}, but it should be s_{1} ≤ s_{0}, if C is contained in the union of the three subgroups M_{1}, M_{2}, M_{3}.

So let n = 4. The case q = 2 was handled using a computer (using a permutation representation on the 135 singular points given in MAGMA), as there are more maximal subgroups containing y.

By [5] every subgroup of G containing y is conjugate (under Aut(G)) to some subgroup of type N_{1} (stabilizer of a nonsingular point) or N_{2}^{−} (stabilizer of a nonsingular O_{2}^{−}-space). So let y centralize a 2-dimensional total nonsingular subspace. Let Ω be a set, on which a group G with pointstabilizer M < G acts transitively and faithfully. Then the following equation holds: |C ∩ M|/|C| = |Fix(x)|/|Ω|, where x is an element of the conjugacy class C. By the action of G on the nonsingular points and 2-spaces of minus-type, we can give an upper bound for |Fix(x)|, which gives us an upper bound for |M ∩C|/|C|. We can show, that an element fixes at most q^{5} + q^{2} resp. 1/2(q^{5} + q^{3}) nonsingular points (all nonsigular points of a 6-dimensional subspace of minus-type resp. all nonsigular points of a 6-dimensional space U with RadU 1-dimensional) of one orbit in case q even resp. odd. The total number of nonsingular points in an orbit is 1/(2, q − 1)q^{3}(q^{4} − 1). So we get \frac{|M∩C|}{|C|} ≤ 0.5/(q + 1) for q ≥ 3. We can show, that an element fixes at most n_{1} + n_{2} nonsingular 2-dimensional subspaces of minus-type, where n_{1} = (q^{6} − 1)(q^{5} − 1)/((q^{2} − 1)(q − 1)) is the number of 2-spaces of a 6-dimensional space and n_{2} = (q^{8} − 1)/(q − 1) is the number of points of an 8dimensional projective space. (The total number of fixed spaces is the number of centralized spaces, which are all inside an at most 6-dimensional space plus the number of fixed, but noncentralized spaces, which one can write as < P, P^{x} >, where P is a point of the 8dimensional projective space.) The number of all 2-dimensional nonsingular subspaces of minus-type is 1/2q^{6}(q^{4} − 1)(q^{3} − 1)/(q + 1), so we get |M ∩ C|/|C| ≤ 0.08 for q ≥ 3 in this case. By action of a triality τ, which just permutes the conjugacy classes and the maximal subgroups, we get an upper bound of |C ∩ M^{τ}|/|C| for every maximal subgroup of G containing y.

By the Zsygmondy-prime dividing o(y) we get, that y lies in exactly one maximal subgroup of each of the three conjugacy classes of subgroups isomorphic to N_{2}^{−}. As y fixes exactly q + 1 nonsingular points, there are at least q + 1 maximal subgroups of type N_{1} containing y. Let y = y_{1}y_{2} = y_{2}y_{1}, where y_{1} has order (q + 1)/(2, q + 1) and y_{2} has order q^{2} − q + 1. There is a triality τ, which fixes y_{2}, as there is a G_{2}(q), which is fixed by τ and contains an element of order q^{2} − q + 1 and all subgroups of this order are conjugate in G by the Zsygmondy-prime. But y_{1}^{τ} ≠ y_{1}, as subgroups of order q^{2} − q + 1 in G_{2}(q) = C_{G}(τ) are selfcentralizing. Assume y is contained in subgroups of type N_{1}^{τ} or N_{1}^{τ^{2}}. Then there is a ŷ ∈ C_{G}(y_{2}) with < ŷ >≠< y > and ŷ fixes a nonsingular point. (ŷ = y^{τ^{2}}resp.y^{τ}). But this point must lie in the 2-dimensional subspace fixed by y_{2}. Thus the stabilizer of this point contains y and ŷ, which is impossible, as < y > is selfcentralizing in this pointstabilizer. This gives us, that there are exactly q + 1 maximal subgroups of type N_{1}^{τ^{i}}. By the upper bound for |M ∩ x^{G}|/|x^{G}| given above we get a conjugate x^{g} with G =< x, y^{g} >=< x, y^{g^{−1}} >. □

Proof of the Theorem

By the classification theorem we had only the groups to consider, which were handled in Lemma 1, 3, 5, 6 and 7. □

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Received July 29, 1997