Covering Numbers for Simple Algebraic Groups

Abstract

Let G be a simple algebraic group over an algebraically closed field, and let C be a noncentral conjugacy class of G. The covering number cn(G,C) is defined to be the minimal k such that G = C^k, where C^k = {c₁c₂⋯c_k : c_i ∈ C}. We prove that $cn(G,C) \le c \frac {\dim G}{\dim C}$, where c is an explicit constant (at most 120). Some consequences on the width and generation of simple algebraic groups are given.

1 Introduction

Given a group G, a subset $C \subseteq G$, and a positive integer k, define C^k = {c₁c₂⋯c_k : c_i ∈ C}. The covering number cn(G,C) is defined to be the minimal value of k such that C^k = G, or $\infty$ if no such k exists. Covering numbers have been studied in a variety of contexts, particularly for the case where G is a simple group and C a conjugacy class of G. For example, in the case where G is a finite simple group of Lie type, it is proved in [4] that cn(G,C) ≤ d rank(G) for all nontrivial conjugacy classes C of G, where d is an absolute constant. A similar result for simple algebraic groups is proved in [7, Theorem 2]: if G is a simple algebraic group over an algebraically closed field K, then cn(G,C) ≤ 4rank(G) for all non-central conjugacy classes C of G. For the case G = PGL_n(K) with n ≥ 3, the sharper result cn(G,C) ≤ n (with equality for some classes) was proved in [16].

Obviously the value of cn(G,C) depends on the class C, and it is desirable to have bounds which reflect this, as opposed to the overall bounds mentioned above. Continuing with the discussion of the case where G is a finite simple group and C a nontrivial conjugacy class, the main result of [19] shows that there is an absolute constant c such that $cn(G,C) \le c \frac {\log |G|}{\log |C|}$ for all such G, C. (Obviously $\frac {\log |G|}{\log |C|}$ is a lower bound for the covering number.) In [22], it is suggested that an analogous result should hold for algebraic groups: namely, there should be a constant c such that

$$ cn(G,C) \le c \frac{\dim G}{\dim C} $$

for all non-central conjugacy classes C of simple algebraic groups G. Such a bound is established in [22, Theorem C] for the case where C is a unipotent conjugacy class in good characteristic. In this paper we prove such a result in general. Note that $\frac {\dim G}{\dim C}$ is an obvious lower bound for cn(G,C).

Theorem 1

There is a constant c such that for any simple algebraic group G over an algebraically closed field K, and any non-central conjugacy class C of G, we have

$$ cn(G,C) \le c \frac{\dim G}{\dim C}. $$

Moreover, this holds with c = 120.

In fact, our proof gives smaller values of the constant c in most cases: specifically, we may take

$$ c = \left\{ \begin{array}{ll} 27&\quad \text{for} G \text{of type} A_{l}, \\ 36&\quad \text{for} G \text{of type} C_{l},\\ 120&\quad \text{for} G \text{of type} B_{l}, D_{l},\\ 18&\quad \text{for} G \text{of exceptional type.} \end{array} \right. $$

These are certainly not the lowest possible values of c, but more effort would be required to reduce them substantially.

Theorem 1 follows from [7] for groups of bounded rank (including exceptional groups). For G = SL_n(K), it is a consequence of a more general result in [21]: from Theorem 2.8 together with the proof of Proposition 2.10 of that paper, it follows that if C₁,…,C_k are conjugacy classes of G = SL_n(K) such that $\sum \dim C_{i} \ge 12 \dim G$, then G = C₁⋯C_k. In particular this implies the conclusion of Theorem 1 with c = 12. A version of this result for arbitrary simple algebraic groups G, but with an unspecified constant in place of 12, follows from [13, Theorem 8.11].

Our proof of Theorem 1 is rather short and elementary, and runs along very different lines to those in [13, 21].

We now give some consequences of Theorem 1. The first concerns the involution width of a simple algebraic group G—that is, the minimal positive integer t such that every element of G can be expressed as a product of t or fewer involutions. Continue to let K denote an algebraically closed field.

Corollary 2

Let G be an adjoint simple algebraic group over K. The involution width of G is at most 2c, where c is the constant in Theorem 1.

This is a simple consequence of Theorem 1, together with the fact that there is a conjugacy class of involutions in G of dimension at least $\frac {1}{2}\dim G$ (see for example [18, §4]). Again, the bound of 2c in the corollary is far from best possible—in fact, as shown to us by the referee, using the main theorem of [20] one can quickly prove that the correct bound is 4, as follows. Let V be the variety of involutions in G. If K is the algebraic closure of a finite field, then V⁴ = G by [20]. The same conclusion then follows for K an arbitrary algebraically closed field of prime characteristic, for example by the statement (1) in the proof of [9, Proposition 1.1]. Finally, one can extend this to characteristic zero by a standard compactness argument, along the lines of [9, §7].

The corollary generalizes to elements of arbitrary order:

Corollary 3

Let r ≥ 2 be an integer, and let G be an adjoint simple algebraic group over K such that G contains an element of order r. Then every element of G can be expressed as a product of at most 2c elements of order r, all of which can be chosen from the same conjugacy class of G.

This again follows from Theorem 1, together with [14], which shows that G has a conjugacy class of elements of order r of dimension at least $\frac {1}{2}\dim G$. Again one can substantially reduce the constant 2c in the conclusion using further arguments, but we do not do this here, as our main point in stating these corollaries is to show some consequences of Theorem 1, rather than obtaining best possible bounds.

The next consequence concerns generation properties of simple algebraic groups G over K. We say that a finite set S of elements generates G topologically, if the subgroup 〈S〉 is Zariski dense in G. Of course, if K is algebraic over a finite field, then G is locally finite, hence is not topologically generated by any finite subset. So assume that K is not algebraic over a finite field. For a non-central conjugacy class C of G, define the generation numbergn(G,C) to be the minimal number d such that G can be topologically generated by d elements of C. It is proved in [10, §8] that gn(G,C) ≤ 2rank(G) + 1 for all classes C. Our final corollary provides a bound that reflects the dependence of the generation number on C:

Corollary 4

Let G be a simple algebraic group over K, and assume that K is not algebraic over a finite field. If C is a non-central conjugacy class of G, then

$$ gn(G,C) \le c \frac{\dim G}{\dim C} + 1, $$

where c is the constant in Theorem 1.

This can be quickly justified using Theorem 1, so we do this here. Let x ∈ C. By [8, Theorem 3.3], there is an element y ∈ G such that G is topologically generated by x, y. And by Theorem 1, we can write y = y₁⋯y_m, where each y_i ∈ C and $m \le c \frac {\dim G}{\dim C}$. Thus G is topologically generated by the m + 1 elements x,y₁,…,y_m, all of which lie in C.

Again, the bound c in the conclusion of Corollary 4 can be greatly improved—in fact the result follows from [6] for SL_n(K) with c = 2, and from [1, 2] in general with c = 5.

The rest of the paper is devoted to the proof of Theorem 1. After some preliminaries in Section 2, we present the proof in Section 3.

2 Preliminaries

Throughout the rest of the paper, K denotes an algebraically closed field of characteristic p ≥ 0.

We begin with a brief discussion of unipotent classes in symplectic and orthogonal algebraic groups, taken from Section 3.3.2 and Chapter 4 of [17]. In Sp_2m(K) and SO_2m(K), there is a subgroup GL_m(K) stabilizing a pair of totally singular m-spaces, and we denote by W(m) a unipotent element that acts as a single Jordan block in this subgroup GL_m(K). Also, in each of Sp_n(K) (n even), SO_n(K) (n odd, p≠ 2) and O_n(K) (n even, p = 2), there is a unipotent element that acts as a single Jordan block, which we denote by V (n). Every unipotent element of a symplectic or orthogonal group G = Sp(V ) or SO(V ) is conjugate to an orthogonal sum

$$ \sum\limits_{i} W(m_{i}) + \sum\limits_{j} V(n_{j}), $$

(1)

where for (G,p) = (SO(V ),2), the number of summands V (n_j) is even. The regular unipotent elements u_reg ∈ G are:

The next lemma shows that regular classes have very small covering numbers.

Lemma 2.1

Let G be a simple algebraic group over K, and let C = x^G be a conjugacy class.

• (i) If x is a regular semisimple element, then C³ = G.

• (ii) If x is a regular element, then C⁴ = G.

• (iii) If x is a regular unipotent element and G = SL_n(K), then C² = G.

Proof

(i) This is [22, Lemma 2].

(ii) Let UT be a Borel subgroup of G with unipotent radical U and maximal torus T, and let x = us be regular, with unipotent part u ∈ U and semisimple part s ∈ T. Let U⁻ denote the maximal unipotent subgroup of G opposite to U. By [3, Theorem 2.1], any non-central conjugacy class of G intersects U⁻s²U. Since C contains an open subset of U⁻s and also of sU in its closure, there is an open subset of U⁻s²U in the closure of C². Thus C² is open in G, and so C⁴ = G.

(iii) This follows from [15, Theorem 5]. □

We shall need to extend Lemma 2.1 to some further unipotent classes of orthogonal groups in characteristic 2. For this we define the following notation. As noted above, there is a subgroup X = GL_m(K) of G = SO_2m(K). For β₁,…,β_m ∈ K^∗, let $d_{\beta _{1},\ldots ,\beta _{m}} = \text {diag}(\beta _{1},\ldots ,\beta _{m}) \in X$; on the natural module for G this acts as a diagonal matrix with eigenvalues $\beta _{i}^{\pm 1}$ for 1 ≤ i ≤ m.

Lemma 2.2

Let p = 2 and m = a + b with a ≥ b ≥ 1, and let u = V (2a) + V (2b) ∈ G = SO_2m(K). Let C = u^G. There exists q₀ such that C⁸ contains all regular semisimple elements $d_{\beta _{1},\ldots ,\beta _{m}}$ of G with $\beta _{i} \in \mathbb {F}_{q}$, for any q ≥ q₀.

Proof

Let q = 2^f, and let F_q be a Frobenius endomorphism of G such that the fixed point group $G^{F_{q}} = {{\varOmega }}_{2m}^{+}(q)$. Write $G(q) = G^{F_{q}}$. We can choose the class representative u to lie in G(q). Let C(q) = u^G(q).

We shall use character theory, via the well-known fact that for any d ≥ 2 and any element z ∈ G(q), the number of solutions (x₁,…,x_d) ∈ C(q) ×⋯ × C(q) to the equation x₁⋯x_d = z is

$$ \frac{|C(q)|^{d}}{|G|} \sum\limits_{\chi\in \text{Irr}(G(q))} \frac{\chi(u)^{d} \chi(z^{-1})}{\chi(1)^{d-1}}, $$

(2)

where the sum is over the set of all irreducible characters Irr(G(q)) of G(q). (For example, this follows easily by induction using [11, Theorem 30.4].)

Let z ∈ G(q) be regular semisimple. We shall show that for d = 8 and for sufficiently large q, the dominant term in the sum in (2) comes from the trivial character, proving that the sum is positive. From this it will follow that z ∈ C(q)⁸, as required for the lemma.

In what follows, c₁,c₂,… are positive absolute constants. In order to analyse the sum in (2), we marshall a few facts:

• (a) By [12], χ(1) ≥ c₁q^2m− 3 for all nontrivial χ ∈Irr(G(q)).

• (b) By [5, Theorem 1.1], the total number k(G(q)) of conjugacy classes of G(q) satisfies k(G(q)) ≤ c₂q^m.

• (c) By [17, 6.3, 6.12] we have |C_G(q)(u)|≤ 2q^{a+ 3b− 2} ≤ 2q^2m− 2.

• (d) As z is regular semisimple, we have |C_G(q)(z)|≤ c₃q^m.

From (c) and (d), it follows that for any nontrivial χ ∈Irr(G(q)),

$$ |\chi(u)| \le |C_{G(q)}(u)|^{\frac{1}{2}} \le c_{5}q^{m-1},\quad |\chi(z)| \le |C_{G(q)}(z)|^{\frac{1}{2}} \le c_{6}q^{\frac{m}{2}}. $$

It follows that

$$ \left |\sum\limits_{1\ne \chi \in \text{Irr}(G(q))} \frac{\chi(u)^{8}\chi(z)}{\chi(1)^{7}}\right | \le \frac{c_{7}q^{m}\cdot q^{8(m-1)}\cdot q^{\frac{m}{2}}}{q^{7(2m-3)}} = c_{7}q^{-\frac{9m}{2}+13}. $$

For m ≥ 3 this tends to 0 as $q\rightarrow \infty$; hence for large q, the sum in (2) is positive, showing as noted above that z ∈ C(q)⁸.

Finally, suppose m = 2, so that u = V (2) + V (2) ∈ G = SO₄. Regarding G as SL₂ ⊗ SL₂, we have u = J₂ ⊗ J₂, the tensor product of two Jordan blocks of size 2. Hence from Lemma 2.1 we have C² = G in this case. □

To conclude this section, it is convenient to deal with some low rank cases of Theorem 1 here:

Lemma 2.3

Suppose that G is a simple algebraic group that is either of exceptional type, or of classical type of rank less than 5. Then $cn(G,C) \le 18 \frac {\dim G}{\dim C}$ for all non-central conjugacy classes C of G.

Proof

This is immediate from [7, Theorem 2], which implies that for the groups G in the hypothesis we have cn(G,C) ≤ 18 for all non-central classes C. □

3 Proof of Theorem 1

Let G be a simple algebraic group of rank l over an algebraically closed field K of characteristic p ≥ 0, and let C = x^G be a non-central conjugacy class of G.

In view of Lemma 2.3, we can assume that G is a classical group of rank l ≥ 5. We shall prove Theorem 1 for G of simply connected type, from which it follows for all isogeny types.

Before starting the proof we describe our strategy, very roughly. First, we write x in Jordan form, and collect blocks together into larger blocks, in such a way that we can apply Lemma 2.1 to each block and obtain a supply of semisimple elements in a small power C^r. We then argue that a further suitable power of C^r contains a regular semisimple element, and deduce the theorem using Lemma 2.1(i).

3.1 The Case G = S L _n(K)

We prove Theorem 1 for G = SL_n(K) with the constant c = 27. Let V = V_n(K) be the natural module for G. Let x ∈ G be non-central and C = x^G. Write x in Jordan normal form as

$$ D \oplus {\bigoplus}_{i=1}^{a} \alpha_{i} J_{r_{i}}(1), $$

where D is diagonal (or empty), the α_i are scalars, and each $J_{r_{i}}(1)$ is a unipotent Jordan block of size r_i ≥ 2. Let

$$ D = {\bigoplus}_{i=1}^{b} \lambda_{i} I_{n_{i}}, $$

where the scalars λ₁,…,λ_b are distinct and n₁ ≥⋯ ≥ n_b. (In the case where D is empty, we treat all n_i as 0 in the following discussion.) For 1 ≤ i ≤ b set D_i = diag(λ₁,…,λ_i). Then x is conjugate to

$$ {\bigoplus}_{i=1}^{a} \alpha_{i} J_{r_{i}}(1) \oplus D_{b}^{n_{b}} \oplus D_{b-1}^{n_{b-1}-n_{b}} \oplus {\cdots} \oplus D_{2}^{n_{2}-n_{3}} \oplus \lambda_{1} I_{n_{1}-n_{2}}. $$

We can conjugate this by elements of the subgroup $SL_{r_{1}} \times {\cdots } \times SL_{2}^{n_{2}-n_{3}}$ and use Lemma 2.1(i,iii) to see that C³ contains diagonal matrices

$$ \text{diag}\left( \beta_{1},{\ldots} ,\beta_{s},{\lambda_{1}^{3}}I_{n_{1}-n_{2}}\right), $$

where s = n − n₁ + n₂ and the β_i are arbitrary, subject to the determinantal conditions $\beta _{1}{\cdots } \beta _{r_{1}} = \alpha _{1}^{3r_{1}}$, etc. There is a product of $\lceil \frac {n}{s} \rceil$ such diagonal matrices that is regular semisimple in G. Hence by Lemma 2.1(i) we have

$$ C^{9\lceil \frac{n}{s} \rceil} = G. $$

(3)

Now C_G(x) contains $GL_{n_{1}}$, so $\dim C \le n^{2}-{n_{1}^{2}}-1$ and

$$ \frac{\dim G}{\dim C} \ge \frac{n^{2}-1}{n^{2}-{n_{1}^{2}}-1}. $$

Also s ≥ n − n₁, and a routine calculation shows that

$$ \frac{n}{n-n_{1}}+1 \le \frac{3(n^{2}-1)}{n^{2}-{n_{1}^{2}}-1}. $$

Therefore $\lceil \frac {n}{s} \rceil \le 3 \frac {\dim G}{\dim C}$, and it follows from (3) that

$$ cn(G,C) \le 27 \frac{\dim G}{\dim C}, $$

as required.

3.2 The Case G = S p _2m(K)

Here we prove Theorem 1 for G = Sp(V ) = Sp_2m(K) with the constant c = 36. Let x ∈ G be non-central and C = x^G. Write x = su, where s, u are the semisimple and unipotent parts of x. Then with respect to a suitable basis, we have

$$ s = I_{2a} \oplus -I_{2b} \oplus \bigoplus_{i=1}^{k} \left( \lambda_{i} I_{c_{i}}, \lambda_{i}^{-1}I_{c_{i}}\right) $$

and

$$ C_{G}(s) = Sp_{2a} \times Sp_{2b} \times \prod\limits_{i=1}^{k} GL_{c_{i}} \le G, $$

where $a+b+\sum c_{i} = m$ and λ_i≠ ± 1 for all i (and of course b = 0 if p = 2). Now u is a unipotent element in C_G(s). Its projections to the factors Sp_2a and Sp_2b are as in (1): letting V_2a,V_2b be the corresponding eigenspaces of s, write

$$ \begin{array}{@{}rcl@{}} V_{2a} \downarrow u &=& W(1)^{c} + \sum W(m_{i}) + \sum V(n_{j}), \\ V_{2b} \downarrow u &=& W(1)^{d} + \sum W(m_{i}^{\prime}) + \sum V\left( n_{j}^{\prime}\right), \end{array} $$

where all $m_{i},m_{i}^{\prime } \ge 2$ and all n_j, $n_{j}^{\prime }$ are even. Now consider the projection of u to a factor $GL_{c_{i}}$ of C_G(s). Write this as a sum of Jordan blocks $J_{1}(1)^{e} + {\sum }_{r_{j}\ge 2}J_{r_{j}}(1)$. Then each block $J_{r_{j}}(1)$ is regular unipotent in a subgroup $GL_{r_{j}}$; and for each block J₁(1) ∈ GL₁ < Sp₂, x acts on the corresponding 2-space as $\left (\lambda _{i},\lambda _{i}^{-1}\right )$, which is regular semisimple in Sp₂. From all this we conclude that

$$ x = I_{2c} \oplus -I_{2d} \oplus \sum\limits_{i} x_{i}, $$

(4)

where each x_i is one of:

• (a) a regular unipotent element of a subgroup $Sp_{2s_{i}}$ (or the negative of such a unipotent);

• (b) a scalar multiple of a regular unipotent element of a subgroup $GL_{s_{i}}< Sp_{2s_{i}}$, s_i ≥ 2;

• (c) a regular semisimple element of a subgroup Sp₂,

and all the relevant subgroups $Sp_{2s_{i}}$, Sp₂ commute with each other.

For $x_{i} \in G_{i} =Sp_{2s_{i}}$ as in (a) or (c), Lemma 2.1 gives ${C_{i}^{4}} = G_{i}$, where C_i is the class $x_{i}^{G_{i}}$. Now consider $x_{i} \in G_{i} =Sp_{2s_{i}}$ as in (b). By Lemma 2.1(iii), ${C_{i}^{2}}$ contains the set of all elements of $GL_{s_{i}}$ of determinant equal to $(\det (x_{i}))^{2}$. This set contains regular semisimple elements of $Sp_{2s_{i}}$ unless s_i = 2 and $\det (x_{i}) = \pm 1$. In the latter case ${C_{i}^{4}}$ contains regular semisimple elements of G_i = Sp₄. We conclude that in each of (a), (b) and (c), ${C_{i}^{4}}$ contains all regular semisimple elements of $Sp_{2s_{i}}$ (satisfying the above determinantal condition in case (b) if s_i ≥ 3).

We need to further refine the decomposition (4) when p≠ 2. Replacing x by − x if necessary, we can assume that c ≥ d. Let y = (I₂,−I₂) ∈ Sp₄, and write x as

$$ I_{2c-2d} \oplus y^{(d)} \oplus \sum\limits_{i} x_{i}, $$

where y^(d) is a block diagonal sum of d copies of y. A matrix computation shows that $(y^{Sp_{4}})^{4}$ contains regular semisimple elements. (Of course if p = 2 then d = 0 and this step is not required.)

At this point it follows that C⁴ contains diagonal matrices

$$ \text{diag}\left( I_{2c-2d},\beta_{1},\beta_{1}^{-1},{\ldots} ,\beta_{s},\beta_{s}^{-1}\right), $$

where s = m − c + d, and the β_i are arbitrary subject to the relevant determinantal conditions. There is a product of $\lceil \frac {m}{s} \rceil$ such diagonal matrices that is regular semisimple in G. Hence by Lemma 2.1(i) we have

$$ C^{12\lceil \frac{m}{s} \rceil} = G. $$

(5)

Now C_G(x) contains Sp_2c, so $\dim C \le 2m^{2}+m-2c^{2}-c$ and

$$ \frac{\dim G}{\dim C} \ge \frac{2m^{2}+m}{2m^{2}+m-2c^{2}-c}. $$

Also s ≥ m − c, and one checks that

$$ \frac{m}{m-c}+1 \le \frac{3(2m^{2}+m)}{2m^{2}+m-2c^{2}-c}. $$

Therefore $\lceil \frac {m}{s} \rceil \le 3 \frac {\dim G}{\dim C}$, and it follows from (5) that

$$ cn(G,C) \le 36 \frac{\dim G}{\dim C}, $$

as required.

3.3 The Case Where G is Orthogonal

Here we prove Theorem 1 for G orthogonal, with the constant c = 120. The simply connected version is a spin group $\hat {G} = Spin_{n}(K)$. However, it is more convenient for us to work with the orthogonal group G = SO_n(K). As above, our method of proof is to show that for any non-central class C of G, some suitable power C^r contains a regular semisimple element of G, hence C^3r = G by Lemma 2.1. If $\hat {C}$ is any class of $\hat {G}$ that maps to C, then $\hat {C}^{r}$ also contains a regular semisimple element of $\hat {G}$, hence $\hat {C}^{3r} = \hat {G}$. Thus our proof of Theorem 1 for G will also give the result for $\hat {G}$.

Assume then that G = SO(V ) = SO_n(K). The proof is quite similar to the symplectic case, but there are a number of extra complications at various points. It is convenient to deal separately with the cases where p≠ 2 and p = 2.

3.3.1 The Case p≠ 2

Suppose first that p≠ 2. Let x ∈ G be non-central and C = x^G. Write x = su, where s,u are the semisimple and unipotent parts of x. With respect to a suitable basis, we have

$$ s = I_{a} \oplus -I_{b} \oplus \bigoplus_{i=1}^{k} \left( \lambda_{i} I_{c_{i}}, \lambda_{i}^{-1}I_{c_{i}}\right) $$

and

$$ C_{G}(s) = \left( O_{a} \times O_{b} \times \prod\limits_{i=1}^{k} GL_{c_{i}}\right)\cap G, $$

where $a+b+2\sum c_{i} = n$ and λ_i≠ ± 1 for all i. As in the symplectic case, letting V_a,V_b be the ± 1-eigenspaces of s, write

$$ \begin{array}{@{}rcl@{}} V_{a} \downarrow u &=& V(1)^{c} + \sum W(m_{i}) + \sum V(n_{j}), \\ V_{b} \downarrow u &=& V(1)^{d} + \sum W(m_{i}^{\prime}) + \sum V\left( n_{j}^{\prime}\right), \end{array} $$

where all $m_{i},m_{i}^{\prime } \ge 2$, and all $n_{j},n_{j}^{\prime }$ are odd and ≥ 3; also d and the number of blocks $V\left (n_{j}^{\prime }\right )$ are either both even or both odd (as $\det (x)=1$). Let the projection of u to each factor $GL_{c_{i}}$ be a sum of e_i Jordan blocks of size 1 and the rest of size ≥ 2. Thus

$$ x = D \oplus \sum\limits_{j} V(n_{j}) \oplus \sum\limits_{j} -V\left( n_{j}^{\prime}\right) \oplus \sum\limits_{i}x_{i}, $$

(6)

where

$$ D = I_{c} \oplus -I_{d} \oplus W(2)^{a} \oplus (-W(2))^{b} \oplus {\sum}_{i} \left( \lambda_{i}I_{e_{i}},\lambda_{i}^{-1}I_{e_{i}}\right), $$

(7)

and each x_i is a scalar multiple (by some λ_i≠ ± 1) of a regular unipotent element of a subgroup $GL_{s_{i}}$, s_i ≥ 2. (Note that we have singled out the ± W(2) summands in D, as W(2) lies in one of the SL₂ factors in SL₂ ⊗ SL₂ = SO₄, so does not have finite covering number in SO₄.)

As in the symplectic case, observe that for each summand $x_{i} \in GL_{s_{i}} < G_{i} = SO_{2s_{i}}$ in (6), $\left (x_{i}^{G_{i}}\right )^{4}$ contains regular semisimple elements of G_i; and for each summand y_j = ±V (n_j), and $G_{j} = SO_{n_{j}}$, we have $\left (y_{j}^{G_{j}}\right )^{4} = G_{j}$.

Next, we need to refine the decomposition of D in (7). We seek to combine all the terms ± W(2), $\left (\lambda _{i},\lambda _{i}^{-1}\right )$, using also ± 1’s from I_c ⊕−I_d, to form a block diagonal sum of non-central elements of SO₆ or SO₈; these can for example take the form W(2)², $\left (W(2),\lambda _{1},\lambda _{1}^{-1}\right )$ or $\left (I_{2},\lambda _{1},\lambda _{2},\lambda _{1}^{-1},\lambda _{2}^{-1}\right )$, and so on. For each such element z_i ∈ G_i = SO₆ or SO₈, we know $\left (z_{i}^{G_{i}}\right )^{10} = G_{i}$, by [7, Theorem 2]. We may also need a block of the form t = (± 1,±W(2)), $\left (\pm 1,\lambda _{1},\lambda _{2},\lambda _{1}^{-1},\lambda _{2}^{-1}\right )$ or $\left (\pm 1,\lambda _{1},\lambda _{1}^{-1}\right )$ in O₅ or O₃; the class of such a block has 8th power containing SO₅ or SO₃.

Assuming that (c,d)≠(0,0) (we postpone the (0,0) case until later), the process in the preceding paragraph can be carried out to write

$$ D = I_{c^{\prime}}\oplus -I_{d^{\prime}} \oplus \sum z_{i} (\oplus t), $$

(8)

where $c^{\prime }\le c, d^{\prime }\le d$. We can refine the $I_{c^{\prime }}\oplus -I_{d^{\prime }}$ part of this further: if $c^{\prime }\ge d^{\prime }$, we can write it as a block sum of a block $B = I_{c^{\prime \prime }}$ ($c^{\prime \prime }\le c^{\prime }$) or (1,− 1), blocks of the form w_i = (I₂,−I₄) or (I₄,−I₂) ∈ SO₆, and at most one block $t^{\prime }=\pm (1,-I_{2}) \in O_{3}$; and if $c^{\prime }< d^{\prime }$ we can similarly write it as a sum of $B = -I_{d^{\prime \prime }}$ ($d^{\prime \prime }\le d^{\prime }$) or (1,− 1), blocks w_i, and at most one further block $t^{\prime }$.

To summarise, we now have

$$ D = B \oplus \sum w_{i} \oplus \sum z_{i} \left( \oplus t \oplus t^{\prime}\right), $$

and so by (6),

$$ x = B \oplus \sum w_{i} \oplus \sum z_{i} \oplus \sum x_{i}\oplus \sum V(n_{j}) \oplus \sum -V\left( n_{j}^{\prime}\right) (\oplus t \oplus t^{\prime}). $$

(9)

In (9), the terms w_i, z_i, x_i lie in orthogonal subgroups of even dimension, and the terms V (n_j), $-V\left (n_{j}^{\prime }\right )$ (and t, $t^{\prime }$) lie in orthogonal subgroups of odd dimension (at least 3). Let c₀ be the total number of these odd terms. Assume $B = I_{c^{\prime \prime }}$ (the other case (1,− 1) is similar and easier). Then from all of the above arguments, we see that C¹⁰ contains diagonal matrices

$$ \text{diag}\left( I_{c^{\prime\prime}+c_{0}},\beta_{1},\beta_{1}^{-1},\ldots,\beta_{s},\beta_{s}^{-1}\right), $$

where $2s = n-c^{\prime \prime }-c_{0}$, and the β_i are arbitrary subject to the relevant determinantal conditions. There is a product of $\lceil \frac {n}{2s} \rceil$ such diagonal matrices that is regular semisimple in G. Hence by Lemma 2.1(i) we have

$$ C^{30\lceil \frac{n}{2s} \rceil} = G. $$

(10)

Now $c^{\prime \prime }+c_{0} \le c + \frac {n-c}{3}$, and hence $2s \ge \frac {2(n-c)}{3}$. Also C_G(x) contains SO_c, so

$$ \frac{\dim G}{\dim C} \ge \frac{n^{2}-n}{n^{2}-n-c^{2}+c}. $$

One checks that

$$ \frac{3n}{2(n-c)}+1 \le \frac{4(n^{2}-n)}{n^{2}-n-c^{2}+c}. $$

Therefore $\lceil \frac {n}{2s} \rceil \le 4 \frac {\dim G}{\dim C}$, and it follows from (10) that

$$ cn(G,C) \le 120 \frac{\dim G}{\dim C}, $$

as required.

It remains to handle the postponed case, where (c,d) = (0,0). In this case, if D cannot be written as in (8), then it must be of one of the following forms, where as before, each z_i ∈ SO₆ or SO₈:

$$ \begin{array}{@{}rcl@{}} \pm W(2) &\oplus& \sum z_{i}, \\ \left( \lambda_{1},\lambda_{2},\lambda_{1}^{-1},\lambda_{2}^{-1}\right) &\oplus& \sum z_{i},\\ \left( \lambda_{1},\lambda_{1}^{-1}\right) &\oplus& \sum z_{i}. \end{array} $$

(11)

Let f = 4 or 2 be the size of the first summand. Then C¹⁰ contains diagonal matrices

$$ \text{diag}\left( I_{f+c_{0}},\beta_{1},\beta_{1}^{-1},\ldots,\beta_{s},\beta_{s}^{-1}\right), $$

where β_i are as above, and c₀ is the number of blocks V (n_j), $-V\left (n_{j}^{\prime }\right )$ in (6). Then $c_{0} \le \frac {n-f}{3}$, so $c_{0}+f \le \frac {n+2f}{3}$. As n ≥ 10 (recall that we are assuming rank(G) ≥ 5), this is less than $\frac {2n}{3}$, and hence there is a product of three such diagonal matrices that is regular semisimple in G. Therefore by Lemma 2.1, we have C⁹⁰ = G.

This completes the proof in the case where p≠ 2.

3.3.2 The Case p = 2

This case follows along very similar lines. Assume that p = 2. As in the previous subsection, write x = su, where

$$ s = I_{a} \oplus \bigoplus_{i=1}^{k} \left( \lambda_{i} I_{c_{i}}, \lambda_{i}^{-1}I_{c_{i}}\right),\quad C_{G}(s) = SO_{a} \times \prod GL_{c_{i}}. $$

Let V_a denote the 1-eigenspace of s. By (1), we can write

$$ V_{a}\downarrow u = \sum W(m_{i}) + \sum V(n_{j}), $$

where all the n_j are even, and the number of summands V (n_j) is even. Again let the projection of u to each factor $GL_{n_{i}}$ be a sum of e_i Jordan blocks of size 1 and the rest of size ≥ 2. Then

$$ x = D \oplus \sum V(n_{j}) \oplus \sum x_{i}, $$

where

$$ D = I_{c} \oplus W(2)^{a} \oplus \sum\limits_{i} \left( \lambda_{i} I_{e_{i}}, \lambda_{i}^{-1}I_{e_{i}}\right) $$

and each x_i is a scalar multiple (by some λ_i≠ 1) of a regular unipotent element of a subgroup $GL_{s_{i}}$, s_i ≥ 2.

Next, we refine the decomposition of D just as in the p≠ 2 case: assuming c > 2, we can write

$$ D = I_{c^{\prime}} \oplus \sum z_{i}, $$

(12)

where $c^{\prime }\le c$ and each summand z_i is non-central in the corresponding orthogonal subgroup SO₆ or SO₈. Thus

$$ x = I_{c^{\prime}} \oplus \sum z_{i} \oplus \sum x_{i} \oplus \sum V(n_{j}). $$

Now using Lemma 2.2, we see that C¹⁰ contains diagonal matrices

$$ \text{diag}\left( I_{c^{\prime}},\beta_{1},\beta_{1}^{-1},\ldots,\beta_{s},\beta_{s}^{-1}\right), $$

where $2s = n-c^{\prime }$, and the β_i ∈ F_q, any q ≥ q₀, are arbitrary subject to the relevant determinantal conditions. As in the previous subsection (see (10)), it follows that

$$ C^{30\lceil \frac{n}{2s} \rceil} = G. $$

Now $2s =n-c^{\prime } \ge n-c$, while $\frac {\dim G}{\dim C} \ge \frac {n^{2}-n}{n^{2}-n-c^{2}+c}$. We check that

$$ \frac{n}{n-c}+1 \le \frac{3(n^{2}-n)}{n^{2}-n-c^{2}+c} $$

and hence $\lceil \frac {n}{2s} \rceil \le 3 \frac {\dim G}{\dim C}$. It follows that $cn(G,C) \le 90 \frac {\dim G}{\dim C}$, giving the result.

Finally, for the case where c ≤ 2, either D can be written in the form (12), or it has one of the forms in (11), or it has the form $(I_{2},\lambda _{1},\lambda _{1}^{-1}) \oplus \sum z_{i}$. Now the conclusion follows by a very similar argument to the one given after (11).

The proof of Theorem 1 is now complete.