Nilpotent covers and non-nilpotent subsets of finite groups of Lie type
Let be a finite group and an element of . A subgroup of is said to be -nilpotent if it is nilpotent and has nilpotency class at most . A subset of is said to be non--nilpotent if it contains no two elements and such that the subgroup is -nilpotent. In this paper we study the quantity , defined to be the size of the largest non--nilpotent subset of .
In the case that is a finite group of Lie type, we identify covers of by -nilpotent subgroups, and we use these covers to construct large non--nilpotent sets in . We prove that for groups of fixed rank , there exist constants and such that , where is the number of maximal tori in .
In the case of groups with twisted rank , we provide exact formulae for for all . If we write for the level of the Frobenius endomorphism associated with and assume that , then may be expressed as a polynomial in with coefficients in .
Key words and phrases:Non-nilpotent set, nilpotent subgroup, nilpotent cover, regular semisimple element, regular unipotent element, finite simple group of Lie type
2010 Mathematics Subject Classification:20D60, 20E07, 20G40
1. Introduction and results
Let be a group and an element of . For , we define to be -nilpotent if is nilpotent of class at most . We define to be -nilpotent if is nilpotent.
1.1. Non--nilpotent subsets
A subset of is said to be non--nilpotent if, for any two distinct elements and in , the subgroup of which they generate is not -nilpotent. The subset is said to be non-nilpotent if it is non--nilpotent.
Define to be the maximum order of a non--nilpotent subset of . It is a trivial observation that for any group , we have
Furthermore, if the class of a nilpotent subgroup of is bounded above by an integer , then clearly
Certainly such a bound exists whenever is finite. Furthermore if is a nilpotent group of class , then for , and for .
Our main interest in this paper will be the quantity , where is a finite group of Lie type. However we shall also consider for various finite values of . In particular the case when is of interest, since is the maximum order of a non-commuting subset (i.e. a subset of such that for all ).
1.2. Nilpotent covers
There is a close connection between the non--nilpotent subsets of maximum order in a group , and -nilpotent covers of . Let be a family of -nilpotent subgroups of . We shall be interested in two possible properties of :
Covering: If for every there exists such that , then we say that is a -nilpotent cover of , or that covers . If , we say that is a nilpotent cover of .
-minimality: If for every subgroup there is an element (called a distinguished element) such that implies that is not -nilpotent, then we say that is -minimal.
We remark that every group admits a -nilpotent cover for all , since the family consisting of all cyclic subgroups is one such. But not every group admits a -minimal -nilpotent cover: examples are the symmetric groups for in the case ([13, 14]).
Suppose that is a -minimal -nilpotent cover, and that contain distinguished elements and respectively. We note that if , then (otherwise the subgroup would be a subgroup of , and hence -nilpotent; this contradicts the condition for -minimality). It follows that the removal from of any one of its members results in a family which is not a cover of (since the distinguished element of the removed member is not in any other member). Hence a -minimal -nilpotent cover of is, in particular, a minimal -nilpotent cover. The converse is false, however: it is not necessarily true that a minimal -nilpotent cover of is -minimal. This is clear from the fact that there exist finite groups with no -minimal -nilpotent covers.
The significance of these properties to the calculation of , and in particular the value of finding a -minimal cover of , is shown by the following proposition.
Let be a family of -nilpotent subgroups of .
If covers , then .
If is -minimal, then .
If is a -minimal cover of , then .
For the first part, suppose is a set of size greater than . If covers , then there exist two elements of lying in one member of , and so cannot be non--nilpotent. The second part is obvious, since the distinguished elements of a -minimal family form a non--nilpotent set. The third part of the proposition follows immediately from the other two. ∎
1.3. Results and structure
The paper is structured as follows. In Section 2 we give some background results on linear algebraic groups. In Section 3 we set out the group-theoretic notation we will use throughout the paper, and we prove a number of basic lemmas pertaining to non--nilpotent groups.
In Section 4 we consider the case that where , where is a simple linear algebraic group of rank , and is a Frobenius endomorphism of . Theorem 4.3 offers a general method for producing lower bounds for , by counting the Sylow subgroups of for certain prime divisors of . In the course of proving this theorem, we provide in Lemma 4.1 a generalization and strengthening of the main result of .
In Section 4 we also prove the following theorem, which is the main result of this paper.
For every , there exist constants such that for any simple linear algebraic group of rank , and for any Frobenius endomorphism of , we have
where is the number of -stable maximal tori in .
The number is given by a result of Steinberg (Proposition 2.5 below) as , where is the level of the Frobenius enomorphism , and is a root system for . In fact it would be possible, in the statement of Theorem 1.2, to take instead to be the number of maximal tori in itself; this point is discussed in the proof of Proposition 4.7 below.
It seems reasonable to conjecture that an upper bound of the same form exists not just for but for , and more speculatively, that the dependence on rank can be removed. We may formulate the following conjecture.
There exist absolute constants such that, for any simple linear algebraic group and any Frobenius endomorphism of , we have
where is the number of -stable maximal tori in .
Theorem 1.2 confirms the existence of a rank-dependent constant . In particular the existence of a constant is confirmed for groups of bounded rank, which includes all the exceptional groups. The existence of a non-rank-dependent constant for the groups follows from the main result of ; and we note that the full conjecture is also confirmed in the case of groups of twisted rank by the results of Section 5 of this paper.
In Section 5 we assume that is a finite group of Lie type where is simple and the twisted rank of is . We prove that, for all , the group admits a -minimal -nilpotent cover. Furthermore we construct explicit examples of such covers and we calculate their order, thereby producing exact formulae for in each case.
The nature of these formulae is somewhat remarkable, and we discuss some of their characteristics in Section 6. We suggest a number of questions and conjectures arising from these observations, and from other results in the paper.
1.4. Background results
The value of has been studied for various groups; it has usually been denoted . Endimioni has proved that if a finite group satisfies , then is nilpotent, while if , then is solvable; furthermore these bounds cannot be improved . Tomkinson has shown that if is a finitely generated solvable group such that , then , where is the hypercentre of (). Also, for a finite non-solvable group , it has been proved by the first author and Hassanabadi that satisfies the condition if and only if (see [2, Theorem 1.2]).
In addition to the results in , the computation of for particular classes of groups has recently started to garner attention. In particular, the first author has given lower bounds for () when . In a forthcoming paper by the second and third authors, a nilpotent cover of is constructed which is -minimal when ; this construction will establish the exact value of for almost all values of and (see ), and an upper bound in the remaining cases.
The particular statistic has attracted considerable recent attention, and been calculated for various groups ; much of this work has concentrated on the case of almost simple groups ([1, 4, 5, 13, 14]).
The study of non-commuting sets in a group , including the study of , goes back many years. In 1976, B. H. Neumann famously answered the following question of Erdős from a few years earlier: if all non-commuting sets in a group are finite, does there exist an upper bound for the size of a non-commuting set in (i.e. is finite)? Neumann answered this question affirmatively by showing that if all non-commuting sets in a group are finite, then is finite . Pyber subsequently gave a strong upper bound for , subject to the same condition on ().
Related to this area of study is the problem of calculating, for a finite -generator group , the size of the largest subset such that any pair of elements of generate . Such sets are closely related to covers of by proper subgroups (in just the same way that we have seen that non--nilpotent sets are related to -nilpotent covers). The statistic has been studied for the symmetric and alternating groups in , and for the groups and in .
1.5. Connections to perfect graphs
We note a connection with a generalization of the commuting graph of a finite group . The commuting graph of is the graph whose vertices are the elements of , with an edge joining vertices and if and only if and commute in . (An alternative definition excludes central elements of ; the distinction is unimportant here.) There is an obvious correspondence between the maximal abelian subgroups of , and the maximal cliques in the commuting graph . From this fact it follows that the minimal size of a covering of by abelian subgroups is equal to the clique cover number of , i.e. the minimal number of cliques required to cover its vertices.
Suppose that has a -minimal abelian cover. Then the set of distinguished elements form an independent set in , and it follows that the clique cover number and the independence number (being the maximal size of an independent set of vertices) are the same for . (It is obvious that the clique cover number is at least as big as the independence number.)
A graph is perfect if the clique cover number and the independence number coincide for every induced subgraph. It appears that it is not known which finite groups have perfect commuting graphs.111Peter Cameron has recently discussed this problem on his blog, at http://cameroncounts.wordpress.com/2011/02/01/perfectness-of-commuting-graphs/.
There is an obvious generalization of as follows: for define to be the graph whose vertices are the elements of , with an edge joining vertices and if and only if the subgroup is -nilpotent. We can generalize the earlier observation connecting abelian covers to properties of in the following way.
A finite group has a -minimal -nilpotent cover if and only if the clique cover number and the independence number of coincide.
Suppose that has a -minimal -nilpotent cover . It is clear that the clique cover number can be no larger than . The first implication therefore follows from the observation that the set of distinguished elements in members of forms an independent set of size .
Conversely suppose that is a -nilpotent cover. If is an independent set of cardinality , then each element of must contain a unique element of , and so is -minimal. ∎
Note that Proposition 1.4 implies that a necessary condition for to be perfect is that admits a -minimal -nilpotent cover.
We remark that, when , it is not necessarily true that the maximal cliques in the resulting graph correspond to maximal -nilpotent subgroups of ; for instance, there exist two -nilpotent groups of order for which the graph has maximal cliques of size . Thus the proof of Proposition 1.4 yields some extra information: we can conclude that if has a -minimal -nilpotent cover , then admits a minimal covering by (not necessarily maximal) cliques such that each clique corresponds to a maximal -nilpotent subgroup of .
In a different direction, one might define the graph to be predictable if all maximal cliques in correspond to maximal -nilpotent subgroups of . One can then ask for which groups , and which values of , is the graph predictable. In particular is it true that if is a simple group, then will be predictable for all ?
2. Background on linear algebraic groups
The material in this section is drawn primarily from . Throughout the section, is a connected reductive linear algebraic group over an algebraically closed field of characteristic .
Since is a linear algebraic group, we can write for some integer . An element is then said to be semisimple if is diagonalizable in , and it is said to be unipotent if its only eigenvalue is . It is a fact that the condition for to be semisimple (respectively unipotent) is independent of the embedding of into , and so we can say that is semisimple (respectively unipotent) element without reference to any particular embedding.
A unipotent subgroup of is a closed subgroup, all of whose elements are unipotent. A Borel subgroup of is a maximal connected closed solvable subgroup of . Borel subgroups always exist, and all Borel subgroups of are conjugate [16, p. 16]. We can write , where is the unipotent radical of (i.e. the maximal connected normal unipotent subgroup of ) and is a maximal torus of (and hence of ). Note that the union of all Borel subgroups of is the whole of , and in particular, any semisimple element of lies in a maximal torus of .
The rank of , written , is defined to be the dimension of a maximal torus in . We write for the Weyl group of and recall that if is a maximal torus of , then .
2.1. Regular elements and centralizers
Any unattributed statements in this section can be found in [16, Section 1.14]. We note first that for all , where denotes the centralizer of in . An element of is said to be regular if . If is regular in , then , the connected component of , is commutative.
The key facts for our purposes concerning regular semisimple elements are given by the following result from .
Let be a semisimple group and let be semisimple. Then the following conditions are equivalent.
is a maximal torus in .
is contained in a unique maximal torus in .
Every connected reductive group contains regular unipotent elements and any two are conjugate in . Let be regular and unipotent; then every semisimple element in lies in , the centre of [16, Proposition 5.1.5]. The following result will be useful [16, Proposition 5.1.3].
Let be a connected reductive group and let be unipotent. Then the following conditions on are equivalent.
lies in a unique Borel subgroup of .
The final proposition of this section follows from [22, Theorem 1.12.5].
Let be a simple algebraic group, and let be the Weyl group of . If is a prime divisor of , then divides .
2.2. Groups of Lie type
A Frobenius endomorphism of a reductive linear algebraic group is an algebraic endomorphism of with the property that there is an algebraic group embedding of into , for an algebraically closed field and , and a power of , say , that is equal to the map on induced by a Galois automorphism of the field . Let be the size of the fixed field of . Then the level of is defined to be (see [22, Definition 2.1.9]).
In what follows, we shall usually adopt the convention that is the level of the Frobenius map . When dealing with the Ree groups and in Section 5, however, it will be convenient to take to be the square of the level of (which is in the notation above).
A Frobenius endomorphism is always an automorphism of , considered as an abstract group, though the inverse map need not be algebraic. We write for the fixed points of under . A finite group of Lie type is a group of the form where is and is a Frobenius endomorphism of . We define the rank of to equal the rank of .
The simple groups of Lie type are defined in [22, Definition 2.2.8]; they are subgroups of , where is a simple linear algebraic group and is a Frobenius endomorphism. A consequence of this terminology is that a simple group of Lie type is not necessarily a finite group of Lie type. For the purposes of studying non-nilpotent sets, however, the next result renders the distinction irrelevant (see the discussion following Lemma 3.6).
Let be a simple group of Lie type. Then for some simple linear algebraic group and Frobenius endomorphism .
If is a Frobenius endomorphism of the group , and if is a closed subgroup of , then we write for the set of points of fixed by .
2.3. Borel subgroups and tori in
We have already mentioned Borel subgroups and tori in the group . We extend this notion to the group as in [16, Section 1.18]. Note first that a subgroup of which satisfies is said to be -stable.
A Borel subgroup of is a subgroup of the form , where is an -stable Borel subgroup of . Any two Borel subgroups of are conjugate in ; what is more, if we write as above, then [18, (69.10)] implies that .
A maximal torus of is a subgroup of the form , where is an -stable maximal torus of . An -stable maximal torus of is said to be maximally split if it lies in an -stable Borel subgroup of , and a maximal torus of is said to be maximally split if it has the form for some maximally split torus in . Any two maximally split tori of are conjugate in .
It follows from the Lang-Steinberg theorem that the group contains an -stable Borel subgroup and that any -stable Borel subgroup contains an -stable maximal torus of . Thus contains both a Borel subgroup and a maximally split maximal torus.
The following result of Steinberg gives a precise count for the number of -stable maximal tori in (see [16, Theorem 3.4.1]; we remark that the result as stated there applies also to the Ree and Suzuki groups). Recall that is defined to be the level of the Frobenius endomorphism .
Let be the root system for . The number of -stable maximal tori in is .
2.4. Sylow -subgroups in
We are interested in the Sylow structure of the group , for which we refer to [22, 30]. Note first the elementary facts that an element is unipotent if and only if it has order for some , while is semisimple if and only if does not divide the order of .
In order to study the structure of a Sylow -subgroup, we consider an -stable Borel subgroup of . If we write as above, then since , and are all -stable, we have . Now [22, Theorem 2.3.4] implies that is a Sylow -subgroup of .
We shall require the following fact, which follows from [30, II.5.19].
Let be a prime dividing , with , and suppose that is a Sylow -subgroup of . Then lies inside , where is an -stable maximal torus of . If is coprime to , then lies inside itself (and so, in particular, lies inside , a maximal torus of ).
Let and be positive integers. A primitive prime divisor (also sometimes called a Zsigmondy prime) of is a prime divisor which does not divide for any . A well-known theorem of Zsigmondy  (in part also due to Bang ) states that if and , then there exists a primitive prime divisor of except in the case . A primitive prime divisor exists also when , unless for some .
We shall need the following simple inequality.
Let be prime, and let for some . Let be such that a primitive prime divisor of exists. Then there is a primitive prime divisor of such that .
It is clear from Zsigmondy’s theorem that if a primitive prime divisor of exists, and if , then a primitive prime divisor to the base also exists; that is, a divisor of which does not divide for any . Now Euler’s theorem states that , where is the totient function. It follows that , and hence that . It is easy to check that the result holds also for the exceptional case in Zsigmondy’s theorem, arising when . ∎
2.5. Classical groups
For certain finite groups of Lie type it is the case that is isomorphic to a central quotient, or a central extension, of a group of similarities of a non-degenerate or zero form , on a vector space over a field . These groups are the classical groups of Lie type.
Table 1 lists the relevant families, with the order of the root system , the order of the associated Wey group , the name of the associated isometry group, and the value of . We include restrictions on the values of and in order to reduce the number of groups that are considered more than once. Note that the dimension of the vector space is equal to the subscript on the isometry group name; we refer to this number as the dimension of the group .
In the proof of Theorem 1.2, we shall use properties of particular families of maximal tori in the classical groups . We call these maximal tori distinguished and we describe them here in terms of the associated formed space . We write for the group of similarities that is either a central quotient or central extension of .
Suppose that is of type , , or . Then it is well known (see, for instance ), that contains an irreducible cyclic subgroup. Let be of maximal order amongst such subgroups (a Singer cycle), and let be a generator of . The irreducibility of guarantees that has distinct eigenvalues. Write for the projection (or for a lift) of in . Then is regular, and so by Proposition 2.1 lies in a unique maximal torus of . The conjugates of in are the distinguished tori in this case.
Suppose that is of type . In this case . Let be an anisotropic vector in , and consider the stabilizer of in . The group contains a subgroup isomorphic to which acts faithfully by isometries on . Now, as in the previous paragraph, this subgroup contains a cyclic subgroup that acts irreducibly on . Taking to be of maximal order amongst such subgroups, and defining and as before, we conclude that is regular and lies in a unique maximal torus of . The conjugates of are the distinguished tori in this case.
Suppose that is of type . In this case, again, is a group of isometries. We let be an anisotropic vector in such that is a space of type . Now the analysis proceeds as in the previous paragraph.
Suppose lastly that is of type . In this case we let be an anisotropic subspace of dimension and observe that is a space of type . Now contains a subgroup isomorphic to . We choose to be the direct product of a cyclic group of maximal order in acting irreducibly on and trivially on , and a cyclic group of maximal order in acting irreducibly on and trivially on . Now choose so that the projection onto generates and the projection onto generates . Then has distinct eigenvalues; its projection (or lift) in is regular, and lies in a unique maximal torus of . The conjugates of are the distinguished tori in this case.
The following lemma will be essential.
Let be a classical group of Lie type of dimension , and let be a distinguished torus in .
If is of type , , or , then let be a primitive prime divisor of ,
If is of type or , then let be a primitive prime divisor of ,
If is of type , then let be a primitive prime divisor of ,
provided, in each case, that a primitive prime divisor exists. Then in all cases where is defined, the torus contains an element of order .
Note that in light of the restrictions on listed in Table 1, Zsigmondy’s theorem implies that is defined in all cases except possibly when is of type , or when .
The lemma follows directly from our definition of a distinguished torus, together with the list of cardinalities of Singer cycles found in [8, Table 1]. ∎
3. Lemmas on non-nilpotent sets
In this section is a group and is an element of . We begin with some results which are useful for giving upper and lower bounds for . The first result, which is based on [5, Lemma 2.4], requires us first to extend the definition of to subsets of groups: for , we write for the size of the largest non--nilpotent subset of .
Let be subsets of such that . Then
Suppose that is a non--nilpotent set in . Then clearly for , and the result follows. ∎
Let be subsets of . Suppose that the following hold:
for , is a non--nilpotent subset of of maximum order,
if and , then is not -nilpotent.
Then is a non--nilpotent subset of , and
The fact that is a non--nilpotent subset of follows immediately from our suppositions, and the lower bound stated for is an obvious consequence. ∎
As mentioned in the introduction, Lemmas 3.1 and 3.2 put into a more general form the connection between nilpotent covers and non-nilpotent sets already established in Proposition 1.1. These lemmas have a number of useful special cases: we refer in particular to [3, Lemma 3.2] and [3, Lemma 3.3].
The following result is a slightly refined version of the latter. For a prime , we write for the number of Sylow -subgroups of a finite group .
Let be finite and let be a prime number dividing . Let the distinct Sylow -subgroups of be . Suppose that
Then the set is -minimal, and there exists a non-nilpotent set such that all elements of are -elements, and such that .
Note first that the condition (1) is equivalent to the assertion that contains elements which lie in a unique Sylow -subgroup of . Since the Sylow -subgroups of are conjugate, we see that every Sylow -subgroup of contains an element which lies in no other Sylow -subgroup. Let be the set . Then clearly .
Let and be distinct elements of , and let . Since no -subgroup of contains both and , it follows that the Sylow -subgroups of are not normal in , and so is not nilpotent. We conclude that is a non-nilpotent set, and hence that the family is -minimal. ∎
We will use Lemma 3.3 in association with the following result.
Let be finite and suppose that are distinct prime divisors of . For , let be a non--nilpotent subset of such that the orders of the elements of are powers of . Suppose that for and , we have whenever . Then
The result follows easily from Lemma 3.2, once we have made the following observation: if and are group elements of prime power order for distinct primes, then the subgroup is nilpotent if and only if . ∎
We end with two related results. The first result is a restated (and slightly generalized) version of [3, Lemma 3.1], which we state here without further proof. The second is a useful corollary to the first.
Let be a normal subgroup of and let be a non--nilpotent subset of . Any set of representatives of in is a non--nilpotent subset of .
Let be a central subgroup of . A set is non--nilpotent if and only if the set
is non--nilpotent in and .
Suppose that is non--nilpotent. Suppose that are such that . Then for some , and so is abelian; it follows that , and we conclude that . If is -nilpotent, then is -nilpotent (since is central), and once again we see that . So is non--nilpotent, as required.
Now suppose that is non--nilpotent and . Then is a set of representatives of in and the result follows from Lemma 3.5. ∎
Lemma 3.6 has an important consequence: it tells us that in order to study non--nilpotent sets in a finite simple group of Lie type , it is sufficient to study them in where is any group such that . Now Lemma 2.4 implies that there is a simple linear algebraic group and a Frobenius endomorphism such that ; therefore it is sufficient to study non--nilpotent sets in the group . For instance, to understand non-nilpotency in , we can study non--nilpotent sets in , where . Similarly, studying non--nilpotent sets in will tell us about non--nilpotent sets in , where .
In the reverse direction, Lemma 3.6 implies that a knowledge of non--nilpotent sets in finite simple groups provides information on the non--nilpotent sets in all quasi-simple groups.
4. Some lower bounds
Throughout this section, is a simple linear algebraic group over an algebraically closed field of characteristic , and is a Frobenius endomorphism for .
4.1. Non-nilpotent sets given by Sylow subgroups
Our first result generalizes the main result of .
Let be a conjugacy class of such that elements of are regular unipotent. There is a set such that and is non-nilpotent.
We can construct such an by including one element of from every Sylow -subgroup of . Since regular unipotent elements lie in a unique Borel subgroup of , they lie in a unique Sylow -subgroup of . Now the result follows from Lemma 3.3. ∎
Suppose that is a conjugacy class of regular semisimple elements in . Suppose also that the elements of have order , where is a prime that does not divide , and is a positive integer. Then has a non-nilpotent subset of size .
Since does not divide , any Sylow -subgroup of lies inside a maximal torus of . Since a regular semisimple element lies in a unique maximal torus of (and hence a unique maximal torus of ), it follows that each regular semisimple element of order lies in a unique Sylow -subgroup of . Now the result follows from Lemma 3.3. ∎
Let be maximal tori in , and suppose that they are pairwise non-conjugate. Suppose that for , the torus contains a regular element of order , where is a prime that does not divide . Then
Let . Since does not divide we see that contains a Sylow -subgroup of . Since is regular of order , we know that it lies inside a unique maximal torus of , and hence for . Furthermore, since is regular, we see that for any . In particular for all .
4.2. Rank dependent bounds for
In this section we complete the proof of Theorem 1.2. To prove the lower bound when is a classical group, we will make use of the distinguished tori that we described in detail in Section 2.5. If is an exceptional group of Lie type, then we take a different approach here, along lines indicated by :
A non-trivial torus of the group is said to be sharp if for every non-identity .
The next result is [6, Theorem 3.1].
All of the exceptional simple groups of Lie type except for contain sharp maximal tori.
We shall require two preliminary results.
For every , there exist positive constants and such that if is any simple algebraic group of rank , is any Frobenius endomorphism of , and is any -stable maximal torus of , then
The upper bound on can be proved easily by checking the orders of the finite groups of Lie type. Now consider the problem of bounding .
Recall first that corresponds to an -conjugacy class of the Weyl group . Indeed if is an element of the -conjugacy class, then where is a maximal split torus, and is the corresponding element in the Weyl group . Let , the cocharacter group of algebraic homomorphisms . Then acts on in a natural way: for we have . Since is a Frobenius map, there is a map such that , where is the map induced by the field automorphism .
Now [16, Proposition 3.3.5] implies that Since has finite order, its eigenvalues on are roots of unity. Thus has eigenvalues , and hence
Now it is easy to see that
where and . ∎
Let . There is a constant such that for any simple algebraic group of rank , for any Frobenius endomorphism of , and for any -conjugacy class of maximal tori of , we have , where is the number of -stable maximal tori in .
For notational convenience in what follows, we shall write as a token for an unspecified function of , with the understanding that separate instances of the token may refer to distinct functions.
Recall that, by Proposition 2.5, where is the root system for . If is bounded above by a function of , then and the proposition holds trivially. Now [16, Proposition 3.6.6] implies that, for sufficiently large, all maximal tori of are non-degenerate, i.e. any maximal torus in lies in precisely one maximal torus of . This implies, in particular, that is equal to the number of maximal tori in .
Proof of Theorem 1.2: lower bound.
We begin by establishing the lower bound for . By Proposition 4.7 it suffices to show that
where is some class of maximal tori.
Suppose that is an exceptional group, other than . Then has a class of sharp tori. Let be distinct tori in . Then clearly . But now for any non-identity and we see that is centreless, and hence not nilpotent. It follows immediately that .
Now suppose that is of classical type. Let be the distinguished torus identified in Section 2.5, and let be the dimension of the group . We may assume without loss of generality, that ; thus Lemma 2.8 implies that, provided is not of type , then contains an element of order , where is a primitive prime divisor of or of or of . If is of type , then Theorems 5.2 and 5.4 imply the result, so we exclude this case and assume that contains the given element .
By Lemma 2.7 we may suppose that