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Invariable generation of finite groups of Lie type

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Invariable generation of finite groups of Lie type

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Content Invariable generation of finite groups of Lie type by Eilidh McKemmie A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2020 Acknowledgments Thanks to my advisor Robert Guralnick for giving me a very interesting topic to work on, for his generosity with his time, and for sharing some of his encyclopaedic knowledge of the classication of nite simple groups. Thank you also to the other members of my thesis and qualifying committees Jason Fulman, Stephan Haas, Susan Montgomery, Eric Friedlander and Nick Warner for their insightful comments. Thanks to John Rahmani for the antics and answering my probability questions. Thanks to Anh Le for the laughs and teaching me new perspectives. Thanks to Daniele Garzoni for talking about invariable generation with me. Thanks to Ian and Susan McKemmie for the opportunities they've given me and the postcards they've sent me. For all their support in the last ve years, I would like to thank Julian Aronowitz, Fanhui Xu, Zhanerke Temirgaliyeva, Ujan Gangopadhyay, Irmak Bal ck, Nicolle Gonz alez, Ezgi Kantarc O guz, Can Ozan O guz, Harrison Algra, Zac Wickham, Austin Pollok, Gene Kim, Kyle Stratton, Dan Douglas, Janejila Snider, Ash Hutchison, Rasa Narbut_ e, William Juan, Albert S lawi nski, Yuanzhong Pan, Brian Yen, Xueer Zhuang, Josh Batra, Gunter Malle, Sean Eberhard, Colva Roney-Dougal, Danny Neftin, Lior Bary-Soroker, Jesse Levitt, and Fedor Malikov. Thanks to Wi KBBQ for keeping me fed, and to ABBA for getting me through all that grading. ii Table of Contents Acknowledgments ii List of Tables iv Abstract v 1 Introduction 1 1.1 History of generation problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Invariable generation and applications . . . . . . . . . . . . . . . . . . . . . 1 1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Preliminaries 5 2.1 Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Finite groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Relationship between algebraic group and Weyl group . . . . . . . . . . . . . 10 3 Increasing rank 12 3.1 Irreducible subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Invariant spaces of regular semisimple elements . . . . . . . . . . . . . . . . 14 3.3 Probabilistic invariable generation of G in terms of W . . . . . . . . . . . . . 18 3.4 Invariably generating the Weyl groups . . . . . . . . . . . . . . . . . . . . . 20 3.5 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Fixed rank 21 4.1 Using the Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Normalizers of tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Orders of elements ofM (T ) . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.4 Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bibliography 30 Appendices 34 Appendix A Experiments to guess the C values . . . . . . . . . . . . . . . . . 34 iii List of Tables 2.1 Semisimple algebraic groups and their root systems . . . . . . . . . . . . . . 6 2.2 Examples of nite groups of Lie type with their Dynkin diagrams . . . . . . 9 2.3 Some Weyl groups with examples of G . . . . . . . . . . . . . . . . . . . . . 11 3.1 Invariant spaces of T w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1 Subgroups M2M with W 0 (M) =W 0 . . . . . . . . . . . . . . . . . . . . . 25 4.2 Tori T and S such thatM (T ) is small and does not contain S . . . . . . . 29 4.3 Tori T and S such thatM (T )\M (S) =; . . . . . . . . . . . . . . . . . 29 iv Abstract A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. The probability that four randomly selected elements invariably generate S n is bounded away from zero by an absolute constant for all n, but for three elements, the probability tends to zero as n!1 [EFG17, PPR16]. We prove analogous results for the nite groups of Lie type as the order of the group grows. We use the fact that most elements of nite groups of Lie type are regular semisimple and the correspondence between classes of maximal tori and conjugacy classes in the extended Weyl group, as well as considering maximal subgroups. v Chapter 1 Introduction 1.1 History of generation problems Mathematicians have long been interested in nding generating sets with nice properties because they can be used to identify groups and do computational work. It is well-known [AG84, Ste62] that all nite simple groups can be generated by a set of size 2. In 1882, Netto [Net82] conjectured that a random pair of elements in the symmetric group generates with probability tending to 3 4 as the size of the group approaches innity. Dixon [Dix69] proved the conjecture in 1969, reigniting activity in probabilistic group theory. Dixon's work was continued by Kantor and Lubotzky [KL90] and Liebeck and Shalev [LS95] to show that two random elements generate a nite simple group G with probability approaching 1 as jGj!1. 1.2 Invariable generation and applications Motivated by the problem of computing Galois groups of random polynomials, Dixon [Dix92] began studying random invariable generation of the symmetric group. Denition 1.2.1. LetS be a subset of a groupG, and deneS = Q s2S s G to be the product of conjugacy classes of elements ofS. We call\ s2S hsi the subgroup ofG invariably generated byS. Note thatS invariably generatesG if and only if every tuple inS is a generating tuple for G. 1 In 2016, Pemantle, Peres and Rivin [PPR16] proved that four random elements of the symmetric group S n invariably generate with probability bounded away from zero by an absolute constant for all n. Eberhard, Ford and Green [EFG17] later showed that three random elements of S n in- variably generate with probability vanishing as n!1. The motivation for studying invariable generation comes from computational Galois the- ory. We consider the problem of deciding whether or not the Galois group of a polynomial f(x)2Z[x] of degree n is S n . Our discussion follows that of Pemantle, Peres and Rivin [PPR16]. There are deterministic algorithms (mentioned in [PPR16]) with complexity O(n 40 ). However, the running time is too high to be practical. There is a good Monte Carlo al- gorithm which does not give false positives but can give false negatives. The correctness of the algorithm relies on the Frobenius Density Theorem. Theorem 1.2.2 (Frobenius Density Theorem, [Jan96]). The density of unramied prime numbers p for which f(x) modulo p has factors whose degrees are d 1 ;:::;d k is equal to the density of elements in Gal(f) which correspond to the partition d 1 + +d k =n. Since there are only nitely many ramied primes, we may throw them out. Then, even though we do not know Gal(f), we may uniformly sample r random elements and get their cycle types 1 ;:::; r by randomly picking primes p 1 ;:::;p r and taking i to be the set of degrees of irreducible factors of f modulo p i . 2 Algorithm 1: Is Gal(f) equal to S n ? input: f(x) =f 0 +f 1 x + +f n x n 2Z[x];f n 6= 0 pick some primes p 1 ;:::;p r uniformly at random from [2; maxfjf i jg]; factor f modulo p i to get the cycle types i ; pick representatives g i of cycle type i ; if the g i invariably generate S n then output \YES" else output \NO" end This algorithm will never give a false positive, but can give a false negative if Gal(f) =S n and we picked elements which do not invariably generate. Assume that the Galois group of f is S n , and that we take 4k random primes in the algorithm. Let " > 0 be a lower bound on the probability that four random elements invariably generate S n . Then the probability of getting a false negative is less than (1") k which approaches 0 as k!1. Due to the number theoretic connection, the body of literature on invariable generation has been growing rapidly. For example, Guralnick and Malle [GM12b], and independently Kantor, Lubotzky and Shalev [KLS11] show that every nite simple group is invariably gen- erated by two elements. Kantor, Lubotzky and Shalev [KLS11] showed that the probability that k random elements of a nite simple group invariably generate is also bounded away from 1 for all integers k 1.3 Results We give the minimum number of random elements required to invariably generate a nite group of Lie type with probability bounded away from zero as the order of the group grows large. Together with the results of Pemantle, Peres and Rivin [PPR16] and Eberhard, Ford 3 and Green [EFG17], we almost nish the result for nite simple groups. Let G be a nite group of Lie type and select a 1 ;:::;a l 2 G uniformly at random. Now let I l be the event that the a i invariably generate G. Main Theorem A. Fix a prime power q and let G be one of the nite groups of Lie type SL n (q), SU n (q), Sp 2n (q), SO 2n+1 (q) or SO 2n . Let denote the Lie type of G. Let n vary. (a) There exist integers C which depend only on the Lie type of G and an absolute constant a2 (0; 1) such that for all q>C and all n we haveP(I 4 )>a. (b) lim q!1 lim n!1 P(I 3 ) = 0. Main Theorem B. LetG be one of the nite groups of Lie type over a eld of prime power order q appearing in Table 4.3. Let be the Lie type of G. Fix the rank of G, and let q vary. There exists a constant b2 (0; 1) depending only on the rank such that for all q we haveP(I 2 )b. Main Theorem A appears in [McK19] where it is extended to the nite simple classical groups, and Main Theorem B will appear in [GM]. Corollary. Let G be a nite simple group that is not a classical group with qC . There is an absolute constant c2 (0; 1) such that the probability that four random elements in G invariably generate G is bounded away from zero by c. Proving the result for classical groups with small q would complete this result for nite simple groups. 4 Chapter 2 Preliminaries 2.1 Algebraic groups We will nd it useful to interpret the nite groups of Lie type as coming from algebraic groups over an algebraically closed eldF p of characteristicp. We use Malle and Testerman's book [MT11] as a reference. Denition 2.1.1. A linear algebraic group (which we will also refer to as an algebraic group) is an ane algebraic variety X with a group structure such that the multiplication and inversion operations :XX!X and i :X!X are morphisms of varieties. Example 2.1.2. SL n (F p ) may be seen as an ane algebraic variety because it is the set of zeroes of the ideal I = (det(t ij ) 1) F p [t ij j 1 i;j n]. Matrix multiplication and inversion are polynomials in the t ij and so give the structure of an algebraic group to SL n (F p ). Other examples of algebraic groups are SO n (F p ) and E 6 (F p ). Denition 2.1.3. A morphism of algebraic groups is a homomorphism of groups which is also a morphism of varieties. An isogeny is a surjective morphism of algebraic groups with nite kernel. Denition 2.1.4. A semisimple group is an algebraic group X which is connected as a topological space and which has no non-trivial closed connected solvable normal subgroup. Example 2.1.5. SL n is semisimple. 5 The semisimple groups have been classied by Chevalley [Che04] as follows. Theorem 2.1.6 (Chevalley Classication Theorem). There is a bijection between isomo- sphism classes of root systems and isogeny classes of algebraic groups. Semisimple algebraic groups come in two avours. The classical groups are those with root systems of types A n ;B n ;C n ;D n , each an innite family indexed by n. The exceptional groups are those with root systems of typesF 4 ;E 6 ;E 7 ;E 8 ;G 2 . Root systems can be depicted by Dynkin diagrams, which are listed in Table 2.1 along with examples of associated groups. Table 2.1: Semisimple algebraic groups and their root systems Example X Dynkin diagram A n1 SL n (F p ) B n SO 2n+1 (F p ) C n Sp 2n (F p ) D n SO + 2n (F p ) E 6 E 6 (F p ) E 7 E 7 (F p ) E 8 E 8 (F p ) F 4 F 4 (F p ) G 2 G 2 (F p ) Denition 2.1.7. A torus of an algebraic group is a closed subgroup isomorphic to the group Diag m GL m (F p ) of diagonal matrices in GL m (F p ) for some m. We call m the dimension of the torus. A maximal torus ofX is a torus maximal with respect to inclusion. All maximal tori are conjugate to each other in X. The rank of X is the dimension of a maximal torus in X. 6 Example 2.1.8. Diag n \ SL n (F p ) = Diag n1 is a maximal torus of SL n (F p ), which means SL n (F p ) has rank n 1. Theorem 2.1.9 (see for example [MT11, Theorem 1.7]). An algebraic group X is a closed subgroup of GL n for some n. Denition 2.1.10. Let X be connected. By Theorem 2.1.9, X may be seen as a closed subgroup of GL n for some n. An element s2 X is called semisimple if it is diagonalisable as an element of GL n . This is well-dened by the Jordan decomposition. Let X be semisimple. An element s2X is called regular if whenever it is contained in a maximal torus T , we have T =C X (s) . By [MT11, Corollary 6.11(a)], every semisimple element in a connected algebraic group is contained in some maximal torus. So for a semisimple group X, a regular semisimple elements is contained in a unique maximal torusC X (s) . This torus is-stable whenevers is-xed: C X (s) is-stable because fora2C X (s) we haves(a)s 1 =(sas 1 ) =(a), and so (C X (s) )C X (s) is a connected component which contains 1, so (C X (s) ) =C X (s) . 2.2 Finite groups of Lie type We may obtain the nite groups of Lie type as xed points of certain endomorphisms of algebraic groups as follows. Let p be a prime and let X be an algebraic group overF p . Identify X with a closed subgroup of GL n (F p ) as in Theorem 2.1.9. Denition 2.2.1. A Frobenius endomorphism of X is the map q : X! X; (x ij )7! (x q ij ) for some q a power of p. A map :X!X is called a Steinberg endomorphism ofX if there is somem2N such that m is a Frobenius endomorphism. 7 LetX be a semisimple algebraic group and :X!X a Steinberg endomorphism. The nite group G =X :=fx2Xj(x) =xg is called a nite group of Lie type. In this way, we get all of the nite classical and exceptional groups. The action of onX induces an action of on the root system associated with X. If this action is non-trivial, then X is called twisted. The Lie type of X is the Lie type ofX with the order of in the pre-superscript. The Lie types of nite groups of Lie type are listed in Table 2.2 along with an example of a group of that type. Example 2.2.2. The map q : SL n ! SL n ; (x ij )7! (x q ij ) is a Steinberg endomorphism and gives rise to the nite group SL n (F p ) q =SL n (q) whose Lie type is A n1 . The map :SL n !SL n ; (x ij )7! (x q ij ) tr is a Steinberg map because 2 (x ij ) = (x q 2 ij ). It gives rise to the special unitary group overq 2 ,SL n (F p ) =SU n (q) whose Lie type is 2 A n1 . One useful fact about Steinberg endomorphisms is as follows (see, for example, [MT11, Proposition 25.1]): Theorem 2.2.3 (Lang-Steinberg). Let G be a connected algebraic group with Steinberg en- domorphism . Then the morphism L :G!G;g7!(g)g 1 is surjective. The term maximal torus has a meaning for nite groups of Lie type. Denition 2.2.4. IfTG and there is a-stable maximal torusT 0 ofX such thatT =T 0 then we call T a maximal torus of G. Since X is a subgroup of X, we can consider regular semisimple elements in X , which Fulman and Guralnick [FG03, Lemma 3.5] show are plentiful: Theorem 2.2.5. The proportion of regular semisimple elements in X is greater than 1 5=(q 1). More accurate estimates for each nite classical group are given by Fulman, Neumann and Praeger [FNP05]. 8 Table 2.2: Examples of nite groups of Lie type with their Dynkin diagrams Example G Dynkin diagram with action of A n1 SL n (q) 2 A n1 SU n (q) B n SO 2n+1 (q) 2 B 2 2 B 2 (q 2 );q 2 = 2 2f+1 C n Sp 2n (q) D n SO + 2n (q) 2 D n SO 2n (q) 3 D 4 3 D 4 (q) E 6 E 6 (q) 2 E 6 2 E 6 (q) E 7 E 7 (q) E 8 E 8 (q) F 4 F 4 (q) 2 F 4 2 F 4 (q 2 );q 2 = 2 2f+1 G 2 G 2 (q) 2 G 2 2 G 2 (q 2 );q 2 = 3 2f+1 9 2.3 Weyl groups Let X be an algebraic group, a Steinberg endomorphism, and G := X a nite group of Lie type. Denition 2.3.1. The Weyl groupW ofX isW :=N X (T )=C X (T ) for some maximal torus T of X. The Weyl group is well-dened up to isomorphism. The Weyl group is generated by a set of involutions, one for each base root in the root system of X. Recall that the Steinberg endomorphism acts on the root system of X, which induces an action of on the Weyl group W . Denition 2.3.2. The extended Weyl group of X is W 0 :=Wohi: The Weyl groups of the classical groups are related to symmetric groups. The Weyl group of typeB n is the same as that of typeC n which isC 2 oS n . By [JK81, Section 4.2], the conjugacy classes of C 2 oS n correspond to signed cycle types, that is, (n " 1 1 ;:::;n "r r ) where " i =1 andn 1 ;:::;n r are positive integers which sum ton. The sign of the conjugacy class is Q " i , and the sign of an element is the sign of its conjugacy class. The Weyl groupW (D n ) of type D n is the subgroup of C 2 oS n consisting of positive elements. We will be working with extended Weyl groups of some Lie types which we list in Ta- ble 2.3, along with an example of a group G with that Lie type. Note that for untwisted groups, the Weyl group and extended Weyl group are the same. 2.4 Relationship between algebraic group and Weyl group Theorem 2.2.3 of Lang and Steinberg gives rise to a correspondence between the algebraic group and its Weyl group which will be very useful in transferring results about invariable generation for Weyl groups to nite groups of Lie type. 10 Table 2.3: Some Weyl groups with examples of G Example G W 0 A n1 SL n (q) S n 2 A n1 SU n (q) S n C 2 B n SO 2n+1 (q) C 2 oS n C n Sp 2n (q) C 2 oS n D n SO + 2n (q) W (D n ) 2 D n SO 2n (q) C 2 oS n G 2 G 2 (q) D 12 Theorem 2.4.1 (see for example [MT11, Proposition 25.1]). LetX be a semisimple algebraic group with Weyl groupW . Take a Steinberg endomorphism and letG :=X . LetW 0 be the extended Weyl group with : W 0 ! W the canonical projection. Assume X has a -stable maximal torus T 0 . Then there is a natural bijection ( G-conjugacy classes of -stable maximal tori of X ) $ ( W -conjugacy classes in the coset W: of in W 0 ) More explicitly, for w = g 1 (g)T 0 2 N X (T 0 )=T 0 = W , the W -conjugacy class (w:) W of w:2W 0 corresponds to the G-conjugacy class of maximal tori containing T w =gT 0 g 1 . Then (T w ) =ft2 T 0 j t = w(t)w 1 g, which is a maximal torus of G, and we will also refer to as T w as long as the meaning remains clear. 11 Chapter 3 Increasing rank In this chapter, we will x the order of the eld and let the rank of the group grow. Our aim is to prove the following. Main Theorem A. Fix a prime power q and let G be one of the nite groups of Lie type SL n (q), SU n (q), Sp 2n (q), SO 2n+1 (q) or SO 2n . Let denote the Lie type of G. Let n vary. (a) There exist integers C which depend only on the Lie type of G and an absolute constant a2 (0; 1) such that for all q>C and all n we haveP(I 4 )>a. (b) lim q!1 lim n!1 P(I 3 ) = 0. In Appendix A, we use GAP [GAP19] to guess at the values of C . We will work withG =SL n (q),SU n (q),Sp 2n (q),SO n (q), take the natural representation ofG acting on a nite dimensional vector space, and then deneJ l to be the event that the a i invariably generate an irreducible subgroup of G. Let L l A be the event that l elements picked uniformly at random from S n invariably generate a transitive subgroup of S n . Note that we might equivalently dene L l A as the event that l random elements of S n do not have conjugates which x a common subset of f1;:::;ng. We will need the following results about the invariable generation of the symmetric group. Theorem 3.0.1 (Pemantle, Peres and Rivin [PPR16]). There exists an absolute constant c2 (0; 1) such that for all n we haveP(L 4 A )>a. 12 Theorem 3.0.2 (Eberhard, Ford and Green [EFG17]). lim q!1 lim n!1 P(L 3 A ) = 0. Eberhard, Ford and Green [EFG17] note that these results hold true if we redene L l A to be the event that thew i invariably generateS n , because of the following result of Luczak and Pyber [ LP93] which tells us that if the w i invariably generate a transitive subgroup of S n then with probability 1 they invariably generate A n or S n . Theorem 3.0.3 ( Luczak and Pyber [ LP93]). The proportion of elements in the union of all transitive subgroups of S n which do not contain S n or A n vanishes as n!1 In Section 3.1, we state Fulman and Guralnick's [FG18] classical group analog of the Luczak and Pyber [ LP93] result which allows us to reduce to irreducible subgroups. Then in Section 3.2 we characterize the maximal tori T w dened in Theorem 2.4.1. This leads to the relationship between the J events and L A discussed in Section 3.3 which will allow us to transfer the results for Weyl groups to nite classical groups. Section 3.4 addresses the sign of random elements necessary to prove the theorem for the orthogonal groups. 3.1 Irreducible subgroups To reduce the problem to a question about irreducible subgroups ofG, we state Fulman and Guralnick's [FG18, Theorem 1.7] analog to Theorem 3.0.3. Theorem 3.1.1 ([FG18]). Let Y (G) be the set of all regular semisimple elements of the union of all irreducible subgroups of G not containing the derived subgroup (for q even and G =Sp 2n (q), exclude the subgroups O 2n (q)). Then lim n!1 jY (G)j=jGj = 0. Now letS be the event that the elementsa i are all regular semisimple. By Theorem 2.2.5, lim q!1 P(S) = 1. Theorem 3.1.2. lim n!1 P(I \S)P(J \S) = 0 and lim n!1 P(I { \S)P(J { \S) = 0: 13 except when G =Sp 2n (q) and q is even, when we get lim n!1 P (I \S) 1 2 1l P (J \S) = 0 and lim n!1 P(I { \S) 1 2 1l P J { \S 2 1l = 0: Proof. In all but nitely many cases, the derived subgroup of G is G. Outside the case of Sp 2n (q) in characteristic 2, the result follows from Theorem 3.1.1. If G =Sp 2n (q) and q is even, we get that regular semisimple elements corresponding to positive elements of W are contained in some conjugate of O + 2n (q) and regular semisimple elements corresponding to negative elements ofW are contained in some conjugate ofO 2n (q). Therefore, if N is the event that all elements have the same sign, by Theorem 3.1.1, lim n!1 P (I \S)P J \N { \S = 0; lim n!1 P(I { \S)P (J \N { ) { \S = 0: These are equal to lim n!1 P (I \S) 1 2 1l P (J \S) ; lim n!1 P(I { \S) 1 2 1l P J { \S 2 1l respectively because N is independent of J and S. 3.2 Invariant spaces of regular semisimple elements Since we have reduced the problem to invariably generating an irreducible subgroup of G, and by Theorem 2.2.5 most elements are regular semisimple, we are interested in invariant spaces of regular sesmisimple elements. Observation 3.2.1. Elements of G fail to invariably generate an irreducible subgroup of G 14 Table 3.1: Invariant spaces of T w G w Invariant spaces of T w SL n (q) (n 1 ; ;n r ) For each n i , an irreducible n i -space. SU n (q) (n 1 ; ;n r ) For each odd n i , an irreducible non-degenerate n i -space. For each evenn i , two irreducible totally singular 1 2 n i -spaces which sum to a non-degenerate n i -space. Sp 2n (n " 1 1 ;:::;n "r r ) For eachn + i , two irreducible totally singularn i -spaces which sum to a non-degenerate 2n i -space. For each n i , an irre- ducible non-degenerate 2n i -space. SO 2n+1 (q) (n " 1 1 ;:::;n "r r ) One 1-dimensional space of the same sign as w. For each n + i , two irreducible totally singular n i -spaces which sum to a non-degenerate 2n i -space of plus type. For each n i , one irreducible non-degenerate 2n i -space of minus type. SO + 2n (n " 1 1 ;:::;n "r r ) For eachn + i , two irreducible totally singularn i -spaces which sum to a non-degenerate 2n i -space of plus type. For each n i one irreducible non-degenerate 2n i -space of minus type. SO 2n (n " 1 1 ;:::;n "r r ) For eachn + i , two irreducible totally singularn i -spaces which sum to a non-degenerate 2n i -space of plus type. For each n i , one irreducible non-degenerate 2n i -space of minus type. if and only if they have conjugates which stabilize a common proper subspace. The following proposition tells us it is enough to look at invariant spaces of tori. Proposition 3.2.2. If s2G is regular semisimple and contained in a maximal torus T of G, then s and T have the same invariant subspaces. Proof. The invariant subspaces of the element s are those spanned by eigenvectors. Since tori are abelian, every regular semisimple element inT has the same set of eigenvectors, and the invariant subspaces are those spanned by eigenvectors. In a vector space with a quadratic or alternating bilinear form, dene the radical of a subspace E to be rad(E) =E\E ? . A space E is called non-degenerate if rad(E) = 0 and totally singular if rad(E) =E. Let T be the torus of diagonal matrices in G. Table 3.1 lists the invariant spaces of T w . We show we may restrict our attention to non-degenerate spaces. 15 Lemma 3.2.3. For G = Sp 2n (q); SU n (q); SO n (q), if the a i stabilize a common (up to conju- gacy) proper subspace then they stabilize a non-degenerate common (up to conjugacy) proper subspace with probability tending to 1 as n!1. Proof. LetU be an invariant common (up to conjugacy) non-zero proper subspace of minimal dimension. Now rad(U) is also a common (up to conjugacy) invariant subspace and so either rad(U) = 0 (U is non-degenerate) or rad(U) =U. In the latter case, since thea i are regular semisimple, they act completely reducibly so U ? = UU 0 where U 0 is a non-degenerate common (up to conjugacy) invariant subspace, unless we haveU =U ? (U is maximal totally singular). LetB be the event we have a maximal totally singular common (up to conjugacy) invariant proper subspace and no non-degenerate one. We will show that the probability that B happens tends to zero in the limit as n!1. B only occurs when G =SU 2n (q);Sp 2n (q);SO + 2n (q). For G =SU 2n (q) or when q is odd andG =Sp 2n (q),B occurs when all cycles in allw i have even length, and the probability of this occurring vanishes in the limit by [FG17, Lemma 4.2]. For G =Sp 2n (q) and q is even, or whenG =SO + 2n (q),B occurs when all cycles in allw i are positive, and the probability of this occurring vanishes in the limit by [FG17, Theorem 4.4]. We can see a connection between the invariant spaces of regular semisimple elements and invariant sets of elements of the Weyl group using Theorem 2.4.1. If we assume our random elementsa 1 ;:::;a l 2G are regular semisimple then we getw 1 :;:::;w l :2W 0 such thata i is contained in some conjugate of T w i . In type A, consider W 0 acting onf1;:::;ng, and in types B;C;D consider W 0 acting onf1;:::;ng (though we will show that in most cases we can forget the signs). Note that elements in W 0 fail to invariably generate a transitive subgroup if and only if there are conjugates of them which stabilize a proper common subset. The sign is important when thinking about the orthogonal groups, so we dene a notion of sign for invariant sets of elements of the Weyl groups of types B and D. 16 Denition 3.2.4. Let W 0 be the Weyl group of type B or D. If w2W 0 stabilizes a set S then we may restrict w to S and dene the sign of S under w to be the sign of wj S . If we havew 1 :;:::;w l :2W 0 all stabilizingS and they all have the same sign onS, then we say that S is a common xed signed subset. Let :W 0 !S n be the canonical projection. Proposition 3.2.5. (a) Let G = SL n (q); SU n (q); Sp 2n (q). Let E be the event that the (w i ) stabilize a common proper subset up to conjugacy if and only if thea i stabilize a common proper subspace up to conjugacy. Then lim n!1 P (E) = 1. (b) Let G = SO 2n (q). Let E so be the event that the w i have a common proper invariant signed subset up to conjugacy if and only if the a i stabilize a common proper subspace up to conjugacy. Then lim n!1 P (E so ) = 1. (c) Let G = SO 2n+1 (q). Dene E so as above. Then lim n!1 P (E so ) = 1 2 1l . Proof. (a) By Lemma 3.2.3, with probability tending to 1 as n!1 the a i stabilize a common (up to conjugacy) proper subspace if and only if for some k < 2n the a i each stabilize a non-degenerate k-space. Since we don't have to worry about stabilizing a totally singular n-space, we may project W 0 onto S n . In Table 3.1 we saw that a i stabilizes a proper non-degenerate subspace if and only if w i stabilizes a proper subset. (b) Let G = SO 2n (q). With probability tending to 1 as n!1 by Lemma 3.2.3, the a i stabilize a common (up to conjugacy) proper subspace if and only if for somek< 2n the a i each stabilize a non-degenerate k-space of the same sign. In Table 3.1 we saw that a i stabilizes a proper non-degenerate subspace with sign " if and only if w i stabilizes a proper subset with sign ". (c) For G = SO 2n+1 (q), if all w i have the same sign (which happens with probability 2 1l , independently of whether the elements stabilize a common (up to conjugacy) subspace) 17 then the a i stabilize a common 1-space. Apart from this, the argument is the same as the SO 2n case. 3.3 Probabilistic invariable generation of G in terms of W There is a relationship between the distribution of elements in W 0 and the distribution of corresponding tori in G. Proposition 3.3.1 ([FG17]). Fix w:2 W 0 such that T w contains a regular semisimple element. Let N G be the proportion of elements in G which are regular semisimple and in a conjugate of T w , and N W be the proportion of elements in W 0 : which are W -conjugate to w:. Then N G N W . Moreover, if we x n and let q!1 we get equality. Proof. By Theorem 2.2.5, the proportion of regular semisimple elements is greater than 1 5=(q 1). Since each regular semisimple element is in exactly one maximal torus, we get N G 1 jGj [G : N G (T w )]jT w j with equality in the limit as q!1. We will use the fact that N G (T w )=T w =C W (w), see for example [MT11, Proposition 25.3]. N G 1 jGj [G :N G (T w )]jT w j with equality as q!1 = 1 jGj [G :T w ] [N G (T w ) :T w ] jT w j = 1 jGj jGj jC W (w)j = 1 jWj jWj jC W (w)j = 1 jWj jw W j =N W : 18 Recall thatL A is the event that thew i 2S n do not x a proper common (up to conjugacy) subset. For typeC, we do not need to consider signed subsets, and so we deneL C to be the event that the (w i ) do not x a proper common (up to conjugacy) subset. In types B and D we must consider instead the eventsL B ;L D that thew i do not x a proper common (up to conjugacy) signed subset. LetS be the event thata 1 ;:::;a l are all regular semisimple. The following result describes the relationship between the invariable generation properties of a nite classical group and its Weyl group when we restrict to regular semisimple elements. Theorem 3.3.2. For large enough n, and with equality as q!1, (a) If G =SL n (q);SU n (q) or SO 2n (q), or if q is odd and G =Sp 2n (q) then P(I \S)P(L ) andP(I { \S)P(L { ): (b) If G =SO 2n+1 (q) or if q is even and G =Sp 2n (q) then P(I { \S) 1 2 1l P(L { ) + 2 1l andP(I \S) 1 2 1l P(L ): Proof. Proposition 3.3.1 tells us that the probability of picking an element from any given W -conjugacy class ofW 0 : is at least the same as the probability of picking a corresponding regular semisimple element in G, with equality in the limit as q!1. Then by Proposi- tion 3.2.5, for large enough n,P(J \S)P(L ) andP(J { \S)P(L { ) and the result follows because, as shown in Theorem 3.1.2, we can replace J with I . We will only use the following special cases of this result here. Corollary 3.3.3. For G = SL n (q);Sp 2n (q);SU n (q);SO 2n (q), or G = SO 2n+1 (q) and for large enough n, P(I 3 \S)P(L 3 ) andP(I 4{ \S) 7 8 P(L 4{ ) + 1 8 : 19 3.4 Invariably generating the Weyl groups Theorem 3.4.1. For all types we have lim n!1 P(L 3 ) = 0 and there exists b > 0 such thatP(L 4 )b for all n. Thanks to Sean Eberhard for suggesting the following proof. Proof. For type A, this was proved by Eberhard, Ford and Green [EFG17]. For type C, we may use the result for type A because lim n!1 P(L C )P(L A ) = 0. For =B;D, we must consider signed sets. Note that if we ignore signs we see that lim n!1 P(L l )P(L l A ) 0 for all l, and so for l = 4, our result holds. For l = 3, one must modify only slightly the proof of Eberhard, Ford and Green [EFG17]. They use disjoint intervalsfI i g i and show that for eachi, restricting only to cycles whose length is in I i , three random elements of S n x a common (up to conjugacy) subset with probability bounded away from zero by some constant, say d i > 0. They then sum over the intervals to show this happens with probability tending to 1 as n!1, using the fact the events are independent. Note that we may pick elements in W 0 by picking elements in S n and then assigning signs to each cycle. For eachi, the probability that all three elements x a common (up to conjugacy) set of the same sign is bounded away from zero by d i 4 . As in [EFG17], we can then sum over the independent events to see that with probability tending to 1, three random elements of W 0 x a common (up to conjugacy) proper signed set. 3.5 Proof of Theorem A By Corollary 3.3.3 lim n!1 P(I 4{ \S) 7 8 lim n!1 P(L 4{ ) + 1 8 which implies lim n!1 P(I 4 [ S { ) 7 8 lim n!1 P(L 4 ) and therefore lim n!1 P(I 4 ) 7 8 lim n!1 P(L 4 )P(S { ), and lim n!1 P(I 3 ) lim n!1 P(I 3 \S)+P(S { ) lim n!1 P(L 3 )+P(S { ) = lim n!1 P(S { ). By Theorem 3.4.1, and since the proportion of regular semisimple elements is at least 15=(q1) by Theorem 2.2.5, we get our result. 20 Chapter 4 Fixed rank In this chapter, we will x the rank of the group and let q grow. Our aim is to prove the following. Main Theorem B. Let G be a nite group of Lie type over a eld of prime power order q appearing in Table 4.3. Let be the Lie type of G. Fix the rank of G, and let q vary. There exists a constant b2 (0; 1) depending only on the rank such that for all q we haveP(I 2 )b. LetN :=fNGjN maximalg be the set of maximal subgroups ofG and consider the set of -stable maximal rank subgroups of X M :=fMXjM is -stable of maximal rankg: ForA a set of subgroups of X, denote the connected parts and the corresponding sub- groups of G respectively by A :=fM jM2Ag; A :=fM jM2Ag: ForA a set of subgroups ofG orX, denote the subsets of subgroups which intersect the conjugacy class of x2G or a torus T of G by A(x) :=fM2Ajx2M g for some g2Gg; A(T ) :=fM2AjTM g for some g2Gg: 21 Note thatx;y2G invariably generateG if and only ifN (x)\N (y) =;. We will mostly be interested in the setsN (x),M (x) andM (T ). Note also thatM (x)M(x) , and we have equality almost always: Lemma 4.0.1. lim q!1 jfx2GjM (x) =M(x) gj jGj = 1: Proof. Note thatM (x)6=M(x) if and only if there is some M2M such that x2 M but x = 2M . By [FG03, Proposition 4.2], lim q!1 j[ g2G (MnM ) g j=jGj = 0: Since there are only nitely many conjugacy classes of maximal rank subgroups of G, we may sum over them to get our result. Fulman and Guralnick [FG03] show that in almost all cases we only need to look at subgroups of maximal rank: Lemma 4.0.2. lim q!1 jfx2GjN (x) =M (x)gj jGj = 1: Proof. Note thatN (x)6=M (x) if and only if x2[ M2NnM M. By [FG03, Theorem 3.2], lim q!1 [ M2NnM M =jGj = 0: Now we reduce to a question about maximal tori. Lemma 4.0.3. Let T 0 be a maximal torus of X and T := T 0 , and let t2 T be regular semisimple. ThenM (T ) =M (t). Proof. Since t2 T we know thatM (T )M (t). For the other inclusion, assume M2 M (t) and therefore there is some maximal rank subgroup M 0 of X such that M = M 0 . Since M 0 is a closed connected algebraic group of maximal rank, it has a unique maximal 22 torus containing t which must also be a maximal torus of X and must therefore be T 0 . So TM which means that M2M (T ). Theorem 4.0.4. There exist b;Q > 0 such that if q > Q and T;S are maximal tori of G withM (T )\M (S) =;, jf(t;s)2TSjt;s invariably generate Ggj jGj 2 b: Proof. Sincet ands invariably generateG if and only ifN (t)\N (s) =;, we need to bound jf(t;s)2TSjN (t)\N (s) =;gj jGj 2 : By Lemma 4.0.2, Lemma 4.0.1 and Theorem 2.2.5, for any"> 0 there isQ such that for all q>Q 0 we have jf(t;s)2TSjM (t)6=N (t) orM (s)6=N (s)gj jGj 2 < " 3 ; jf(t;s)2TSjM (t)6=M(t) orM (s)6=M(s) gj jGj 2 < " 3 ; jf(t;s)2TS not both regular semisimplejg jGj 2 < " 3 : Then for q>Q 0 , jf(t;s)2TSjN (t)\N (s) =;gj jGj 2 > jf(t;s)2TSjM (t)\M (s) =;gj jGj 2 " 3 > jf(t;s)2TSjM (t)\M (s) =;gj jGj 2 2 3 ": 23 We may use Lemma 4.0.3 to get jf(t;s)2TSjN (t)\N (s) =;gj jGj 2 > jf(t;s)2TSjM (T )\M (S) =;gj jGj 2 " jf(t;s) both regular semisimple in some conjugate of TSgj jGj 2 ": Now for large enough q, taking w 1 ;w 2 2 W such that T := T w 1 and S := T w 2 , Proposi- tion 3.3.1 tells us jf(t;s)2TSjt;s invariably generate Ggj jGj 2 w W 1 w W 2 jWj 2 ": Settingb = 1 2 min w2W f w W =jWjg 2 and picking"<b, we are able to pickQ large enough such that for all q>Q, we have jf(t;s)2TSjt;s invariably generate Ggj jGj 2 b: Our aim now is to nd -stable maximal tori T 0 and S 0 of X such that, for T :=T 0 and S :=S 0 , we haveM (T )\M (S) =;. 4.1 Using the Weyl group If the extended Weyl groupW 0 ofG has an invariably generating pair, then we can use this information to choose our tori. Theorem 4.1.1. If w 1 :;w 2 :2 W 0 invariably generate a subgroup W 1 W 0 and M 2 M (T w 1 )\M (T w 2 ) with extended Weyl group W 0 (M) then W 1 W 0 (M). Proof. Let M2M (T w 1 )\M (T w 2 ). Then there are G-conjugates of T w 1 ;T w 2 which are 24 maximal tori of M. Consider the Weyl group W (M) and the extended Weyl group W 0 (M) of M which are subgroups of W and W 0 respectively. Since T w i is a torus of M, it corresponds to a W (M)-conjugacy class of W 0 (M): which is a subset ofw i : W , so there areW -conjugates ofw 1 :;w 2 : which are contained inW 0 (M), so by assumption W 1 W 0 (M). Corollary 4.1.2. If w 1 :;w 2 : invariably generate W 0 or some maximal subgroup which is not the extended Weyl group of any member ofM and M 2M (T w 1 )\M (T w 2 ) then W 0 (M) =W 0 . Now let M2M such that W 0 (M) =W 0 . We will list all possibilities for such MG. Because W (M) < W is a proper subgroup, we see that M must be twisted with Steinberg endomorphism M . SincejW 0 (M)j =jW 0 j we havejW (M)jj M j =jWjjj. By writing down all the options, one sees that in all but one case, ifjW 0 (M)j =jW 0 j thenW 0 (M) =W 0 . The extended Weyl groups of 2 G 2 (q) and SL 4 (q) have the same order but are not isomorphic. These results are listed in Table 4.1. Table 4.1: Subgroups M2M with W 0 (M) =W 0 G M SL 2 (q) T (2) G 2 (q) SU 3 (q) Sp 2n (q) SO 2n (q) SO 2n+1 (q) SO 2n (q) Now we ndw 1 :;w 2 :2W 0 which invariably generate an appropriately large subgroup while also avoiding the situations of Table 4.1, giving usM (T w 1 )\M (T w 2 ) =;: A n1 : The extended Weyl group is S n . For n = 2, we must avoid generating only T (2) , so we pick (1 2 ); (2)2S 2 . Now assumen> 2. By Jordan's Theorem, a primitive subgroup of S n containing a prime cycle of lengthn 3 is either A n or S n . A group containing an n-cycle is transitive, and if it contains also a cycle of prime order greater than n=2 then it is primitive. So, for n 8 and for r2 (n=2;n 3], S n is invariably generated 25 byf(n); (1 nr2 ; 2;r)g. We know that such a prime exists for all n 8 since Jitsuro Nagura [Nag52] showed it for all n 52 and we can check the smaller cases with a computer. For n < 8 we note that when n is a prime, S n is invariably generated by f(1 n2 ; 2); (n)g, and that S 4 is invariably generated byf(1; 3); (4)g. Although S 6 has no invariably generating pair, there is a maximal subgroup W 1 of S 6 which is not the extended Weyl group of any member ofM . LetW 1 be the transitive maximal subgroup of S 6 which is isomorphic to S 5 and is invariably generated by a 5-cycle and a 6-cycle. The maximal rank subgroups of SL 6 (q) with Weyl group S 5 all come from subsystems of typeA 4 . These Weyl groups are all intransitive as subgroups of S 6 , and so cannot be W 1 . So we may take w 1 a 5-cycle and w 2 a 6-cycle. For the other classical groups with n = 6, the extended Weyl groups project onto S 6 via : W 0 ! S 6 . The same argument as for SL 6 (q) shows that if a maximal rank subgroup ofG has an extended Weyl groupW 0 0 such that(W 0 0 ) =S 5 then(W 0 0 ) is a transitive subgroup of (W 0 ). Therefore it is enough to invariably generate W 0 1 W 0 maximal such that (W 0 1 ) =W 1 . 2 A n1 : The extended Weyl group is W 0 = S n hi, and we can see that if w 1 ;w 2 invariably generate S n then w 1 :;w 2 : invariably generate W 0 , and if w 1 ;w 2 invariably generate W 1 S 6 then w 1 :;w 2 : invariably generate W 1 hi: B n : W (B n ) = C 2 oS n and let : W (B n )! S n be the canonical projection. If H W (B n ) and (H) = S n then the only possible options for ker are the trivial group, the diagonal subgrouph(;; ;)i, the subgroupW (D n ) of positive elements and W (B n ). For n6= 6, we know there are w 1 ;w 2 2 S n which invariably generate S n . If we pickc 1 ;c 2 2C n 2 thenc 1 :w 1 ;c 2 :w 2 invariably generateW (B n ) if and only ifc 1 ;c 2 are not both diagonal or positive. S n is invariably generated by an n-cycle and another element . Therefore W (B n ) is invariably generated by a positive n-cycle and the same with exactly one cycle 26 negative. For n = 6 take W 0 1 =C 2 oW 1 which is invariably generated by w 1 of type (1 + ; 5 ) and w 2 a positive 6-cycle. Here, we have SO 2n (q)2M (T w 1 ) but not inM (T w 2 ), and soM (T w 1 )\M (T w 2 ) = ;: C n : The extended Weyl group is the same as that of type B n . D n : W (D n ) is the set of positive elements of W (B n ), so we have the canonical projection : W (D n )! S n . If H W (D n ) and (H) = S n then the only possible options for ker are the trivial group, the diagonal subgrouph(;; ;)i, and W (D n ). For n6= 6, we know there are w 1 ;w 2 2 S n which invariably generate S n . If we pick c 1 ;c 2 2 C n 2 then c 1 :w 1 ;c 2 :w 2 invariably generate W (D n ) if and only if c 1 ;c 2 are not both diagonal. S n is invariably generated by an n-cycle and another element . Therefore W (D n ) is invariably generated by a positive n-cycle and the same with exactly two cycles negative. For n = 6 take W 0 1 to be the group of positive elements in C 2 oW 1 which is invariably generated by w 1 of type (1 ; 5 ) and w 2 a positive 6-cycle. 2 D n : The extended Weyl group is the same as that of type B n , and the coset of is the set of negative elements, so we can use the same w 1 and w 2 as for type B n . G 2 : The extended Weyl group of G 2 (q) is D 12 , the dihedral group of order 12. This is invariably generated by a re ection and a rotation. We must pick carefully to avoid generating the SU 3 (q) subgroup, so we pick a re ection whose maximal torus is con- tained instead in the SL 3 (q) subgroup and not SU 3 (q). Aschbacher [Asc87] calls the re ection and the rotation s. 27 4.2 Normalizers of tori For the cases G = E 8 (q), G = 3 D 4 (q), G = 2 F 4 (q 2 );q 2 8, G = 2 B 2 (q 2 );q 2 8, and G = 2 G 2 (q 2 );q 2 27, Weigel [Wei92] exhibits a cyclic torusT such thatM (T ) =fN G (T g )j g2Gg. We list them in Table 4.3 Theorem 4.2.1. The only maximal torus of G contained in N G (T ) is T . Proof. Let T 0 be a maximal torus of X such that T 0 = T . Let U 0 be a maximal torus of X such that U 0 N G (T ). Then U 0 N X (T 0 ) . Since X is semisimple, by [MT11, Theorem 3.10, Corollary 8.13] we get that N X (T 0 ) =T 0 , so U 0 =T 0 . So we pickS to be any maximal torus ofG which is not conjugate toT . Since conjugacy classes of maximal tori correspond to -conjugacy classes in W of which there is more than one, such a torus S exists. 4.3 Orders of elements ofM (T ) In the remaining cases, we can nd maximal toriT;S ofG such that the order ofS does not divide any of the orders of elements ofM (T ), so thatM (T )\M (S) =;. Weigel [Wei92] and Guralnick and Malle [GM12a] identify cyclic maximal tori T such that all elements of M (T ) are isomorphic. Kantor and Seress [KS02] list the orders of maximal tori of G, and we only need to nd S whose order doesn't divide the order of elements inM (T ). Then we haveM (T )\M (S) =;. The results of this are given in Table 4.2: 4.4 Proof of Theorem B For each nite group of Lie typeG, we have given maximal toriT andS such thatM (T )\ M (S) =; in Table 4.3. By Theorem 4.0.4, we are done. 28 Table 4.2: Tori T and S such thatM (T ) is small and does not contain S T is a cyclic maximal torus whose connected maximal rank overgroups are listed. There is a torusS with the given order which by order considerations must not be contained in any of the overgroups of T listed. G T M2M (T ) jMj jSj F 4 (q);q 3 12 3 D 4 (q):3 3q 12 12 2 6 2 3 2 2 2 1 4 1 E 6 (q) 9 =(3; 1 ) SL 3 (q 3 ):3 3q 9 9 6 2 3 2 2 1 6 1 =(3; 1 ) 2 E 6 (q) 18 =(3; 2 ) SU 3 (q 3 ):3 3q 9 18 2 6 3 2 2 1 6 2 =(3; 2 ) E 7 (q) 2 18 =(2; 1 ) 2 E 6 (q) sc :D q+1 2q 36 18 12 10 8 3 6 2 4 2 3 7 2 4 1 7 1 =(2; 1 ) Table 4.3: Tori T and S such thatM (T )\M (S) =; Let r be some prime in (n=2;n 3]. G T S SL 2 (q) T (1 2 ) T (2) SL 4 (q); SU 4 (q) T (1;3) T (4) SL 6 (q); SL 6 (q) T (1;5) T (6) SL r (q); SU r (q);r prime T (1 r2 ;2) T (r) SL n (q); SU n (q);n 8 T (n) T (1 nr2 ;2;r) Sp 8 (q); SO 9 (q); SO 8 (q) T (1 + ;3 ) T (4 + ) Sp 12 (q); SO 13 (q); SO 12 (q) T (1 + ;5 ) T (6 + ) Sp 2r (q); SO 2r+1 ; SO 2n (q);r prime T ((1 + ) r2 ;2 ) T (r + ) Sp 2n (q); SO 2n+1 (q); SO 2n (q);n 8 T (n + ) T ((1 + ) nr2 ;2 + ;r ) SO + 8 (q) T (1 ;3 ) T (4 + ) SO + 12 (q) T (1 ;5 ) T (6 + ) SO + 2r (q);r prime T ((1 + ) r3 ;1 ;2 ) T (r + ) SO + 2n (q);n 8 T (n + ) T ((1 + ) nr2 ;2 ;r ) G 2 (q) T T s F 4 (q);q 3 12 Torus of order 4 1 E 6 (q) 9 =(3; 1 ) Torus of order 6 1 =(3; 1 ) 2 E 6 (q) 18 =(3; 2 ) Torus of order 6 2 =(3; 2 ) E 7 (q) 2 18 =(2; 1 ) Torus of order 7 1 =(2; 1 ) E 8 (q) 30 Any torus not conjugate to T 3 D 4 (q) 12 Any torus not conjugate to T 2 F 4 (q 2 );q 2 8 q 4 + p 2q 3 +q 2 + p 2q + 1 Any torus not conjugate to T 2 B 2 (q 2 );q 2 8 q 2 + p 2q + 1 Any torus not conjugate to T 2 G 2 (q 2 );q 2 27 q 2 + p 3q + 1 Any torus not conjugate to T 29 Bibliography [AG84] Michael Aschbacher and R Guralnick. Some applications of the rst cohomology group. J. Algebra, 90(2):446{460, 1984. [Asc87] Michael Aschbacher. Chevalley groups of type g2 as the group of a trilinear form. J. Algebra, 109(1):193{259, 1987. [Bri02] John R. Britnell. Cyclic, separable and semisimple matrices in the special linear groups over a nite eld. J. London Math. Soc., 66(3):605{622, 2002. [Bri06] John R. Britnell. Cyclic, separable and semisimple transformations in the special unitary groups over a nite eld. J. Group Theory, 9(4):547, 2006. [Che04] Claude Chevalley. Classication des groupes alg ebriques semi-simples: The classi- cation of semi-simple algebraic groups, volume 3. Springer, 2004. [Dix69] John D Dixon. The probability of generating the symmetric group. Math. Z., 110(3):199{205, 1969. [Dix92] John D Dixon. Random sets which invariably generate the symmetric group. Discrete Math., 105(1-3):25{39, 1992. [EFG17] Sean Eberhard, Kevin Ford, and Ben Green. Invariable generation of the symmetric group. Duke Math. J., 166(8):1573{1590, Jun 2017. [FG03] Jason Fulman and Robert Guralnick. Derangements in simple and primitive groups. Groups, combinatorics and geometry, pages 99{121, 2003. 30 [FG17] Jason Fulman and Robert Guralnick. Derangements in subspace actions of nite classical groups. Trans. Amer. Math. Soc., 369(4):2521{2572, 2017. [FG18] Jason Fulman and Robert Guralnick. Derangements in nite classical groups for actions related to extension eld and imprimitive subgroups and the solution of the Boston{Shalev conjecture. Trans. Amer. Math. Soc., 370(7):4601{4622, 2018. [FNP05] Jason Fulman, Peter M. Neumann, and Cheryl E. Praeger. A generating function approach to the enumeration of matrices in classical groups over nite elds. Mem. Amer. Math. Soc., 830, Jul 2005. [GAP19] The GAP Group. GAP { Groups, Algorithms, and Programming, Version 4.10.2, 2019. [GL01] Robert M. Guralnick and Frank L ubeck. On p-singular elements in Chevalley groups in characteristic p. Groups and computation, III (Columbus, OH, 1999), 8:169{182, 2001. [GM] Daniele Garzoni and Eilidh McKemmie. On the probability of generating invariably a nite simple group. In preparation. [GM12a] Robert Guralnick and Gunter Malle. Products of conjugacy classes and xed point spaces. Journal of the American Mathematical Society, 25(1):77{121, 2012. [GM12b] Robert Guralnick and Gunter Malle. Simple groups admit Beauville structures. J. London Math. Soc., 85(3):694{721, 2012. [Jan96] Gerald J Janusz. Algebraic number elds, volume 7. American Mathematical Soc., 1996. [JK81] Gordon James and Adalbert Kerber. The representation theory of the symmetric group. 1981. Encyclopedia Math. Appl, 16, 1981. 31 [KL90] William M Kantor and Alexander Lubotzky. The probability of generating a nite classical group. Geom. Dedicata, 36(1):67{87, 1990. [KLS11] WM Kantor, A Lubotzky, and A Shalev. Invariable generation and the chebotarev invariant of a nite group. J. Algebra, 348(1):302{314, 2011. [KS02] William M Kantor and Akos Seress. Prime power graphs for groups of lie type. J. Algebra, 247(2):370{434, 2002. [ LP93] Tomasz Luczak and L aszl o Pyber. On random generation of the symmetric group. Comb. Probab. Comput., 2(4):505{512, 1993. [LS95] Martin W Liebeck and Aner Shalev. The probability of generating a nite simple group. Geom. Dedicata, 56(1):103{113, 1995. [McK19] Eilidh McKemmie. Invariable generation of nite classical groups. ArXiv e-prints, 2019. [MT11] Gunter Malle and Donna Testerman. Linear algebraic groups and nite groups of Lie type, volume 133. Cambridge University Press, 2011. [Nag52] Jitsuro Nagura. On the interval containing at least one prime number. Proc. Japan Acad., 28(4):177{181, 1952. [Net82] Eugen Netto. Substitutionentheorie und ihre Anwendungen auf die Algebra. BG Teubner, 1882. [PPR16] Robin Pemantle, Yuval Peres, and Igor Rivin. Four random permutations conju- gated by an adversary generate S n with high probability. Random Struct. Algo- rithms, 49(3):409{428, 2016. [Ste62] Robert Steinberg. Generators for simple groups. Can. J. Math., 14:277{283, 1962. 32 [Wei92] Thomas S Weigel. Generation of exceptional groups of lie-type. Geometriae Ded- icata, 41(1):63{87, 1992. 33 Appendix A Experiments to guess the C values We did some computational experiments to make a guess at the value of C . By the proof of Main Theorem A we have P(I 4 ) 7 8 P(L 4 )P(S { ), and there exist (listed below) specic bounds forP(S { ), so we just need to knowP(L 4 ) to give the values of C . We do not have an explicit bound forP(L 4 ), but we can guess it using GAP [GAP19]. The function InvGenTest(n, l, t) picks l elements of S n uniformly at random and tests if there is a common xed set size. It performs t such tests and outputs the proportion of random tuples which invariably generate a transitive subgroup of S n . In order to make a guess atP(L 4 ) we test with l = 4;t = 100 and vary n. gap> InvGenTest(10, 4, 100); 0.67 gap> InvGenTest(100, 4, 100); 0.57 gap> InvGenTest(1000, 4, 100); 0.49 gap> InvGenTest(10000, 4, 100); 0.53 gap> InvGenTest(100000, 4, 100); 0.47 gap> InvGenTest(2,4,100); 34 0.89 gap> InvGenTest(2*3,4,100); 0.76 gap> InvGenTest(2*3*5,4,100); 0.65 gap> InvGenTest(2*3*5*7,4,100); 0.58 gap> InvGenTest(2*3*5*7*11,4,100); 0.6 gap> InvGenTest(2*3*5*7*11*13,4,100); 0.51 gap> InvGenTest(2*3*5*7*11*13*17,4,100); 0.52 gap> InvGenTest(3,4,100); 0.83 gap> InvGenTest(3*5,4,100); 0.77 gap> InvGenTest(3*5*7,4,100); 0.62 gap> InvGenTest(3*5*7*11,4,100); 0.59 gap> InvGenTest(3*5*7*11*13,4,100); 0.55 gap> InvGenTest(3*5*7*11*13*17,4,100); 0.42 This result seems to suggest thatP(L 4 )> 1=3. 35 Theorem A.0.1. The proportion of regular semisimple elements in some specic G are listed in this table. G q Limiting proportion of regular semisimple elements Reference SL n (q) any 1 1=q [Bri02] Sp 2n (q) even 1 2=q + 2=q 2 +O(1=q 3 ) [FNP05] Sp 2n (q) odd 1 3=q + 5=q 2 +O(1=q 3 ) [FNP05] SU n (q) any 1 1=q +O(1=q 3 ) [Bri06] SO () n (q) odd 1 2=(q 1) 1=(q 1) 2 [FG17, GL01] SO 2n+1 (q) even 1 2=q + 2=q 2 +O(1=q 3 ) see below SO 2n (q) even 1 2=q + 2=q 2 +O(1=q 3 ) see below Proof. References are given for most of the groups. The result for SO 2n+1 (q) with q even follows from O 2n+1 (q) =Sp 2n (q). The result for SO 2n (q) for even q comes from the facts that the proportion of regular semisimple elements inO 2n (q) is half that inSp 2n (q) [FNP05], and that all regular semisimple elements in O 2n (q) are in SO 2n (q). If we use the guessed boundP(L 4 )> 1=3 and use the above table of bounds forP(S { ), we get the following guesses for C . G C SL n (q) 12 Sp 2n (q) 36 SU n (q) 12 SO () n (q) 25 Here is the code. OneInvGenTest := function(G, n, l) local sizes, i, r, intersection; 36 sizes:=[]; for i in [1 .. l] do r:=Random(G); Add(sizes, FixedSpaceSizes(r, n)); od; intersection := Intersection(sizes); return intersection=[]; end; InvGenTest:= function(n, l, t) local G, i, total; total := 0; G:=SymmetricGroup(n); for i in [1 .. t] do; if OneInvGenTest(G, n, l) then total := total + 1; fi; od; return Float(total/t); end; 37

Abstract (if available)

Abstract A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. The probability that four randomly selected elements invariably generate Sₙ is bounded away from zero by an absolute constant for all n, but for three elements, the probability tends to zero as n → ∞. We prove analogous results for the finite groups of Lie type as the order of the group grows. ❧ We use the fact that most elements of finite groups of Lie type are regular semisimple and the correspondence of classes of maximal tori and conjugacy classes in the extended Weyl group, as well as considering maximal subgroups.

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Creator McKemmie, Eilidh (author)

Core Title Invariable generation of finite groups of Lie type

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 07/23/2020

Defense Date 04/08/2020

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag classical groups,Generation,group theory,invariable,Lie type,maximal subgroups,maximal torus,OAI-PMH Harvest,probabilistic,random,regular semisimple,Weyl group

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Guralnick, Robert (committee chair), Fulman, Jason (committee member), Haas, Stephan (committee member)

Creator Email eilidh.mckemmie@gmail.com,mckemmie@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-342889

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