On the second-largest Sylow subgroup of a finite simple group of Lie type
aa r X i v : . [ m a t h . G R ] D ec On the second-largest Sylow subgroupof a finite simple group of Lie type
S. P. Glasby, Alice C. Niemeyer, and Tomasz Popiel
Abstract.
Let T be a finite simple group of Lie type in characteristic p , and let S be aSylow subgroup of T with maximal order. It is well known that S is a Sylow p -subgroupexcept in an explicit list of exceptions, and that S is always ‘large’ in the sense that | T | < | S | | T | . One might anticipate that, moreover, the Sylow r -subgroups of T with r = p are usually significantly smaller than S . We verify this hypothesis by provingthat for every T and every prime divisor r of | T | with r = p , the order of the Sylow r -subgroup of T at most | T | ⌊ log r ( ( ℓ + ) r ) ⌋ /ℓ = | T | O ( log r ( ℓ ) /ℓ ) , where ℓ is the Lie rank of T . MSC 2010 Classification: 20D08, 20E32, 20E07
1. Introduction
Given a finite simple group T of Lie type, it is natural to ask: what is the order of thelargest Sylow subgroup of T ? This question dates back at least to 1955 and the articles[ , ] of Artin, who showed that if T is a classical group in characteristic p and S is aSylow subgroup of T with maximal order, then S is a Sylow p -subgroup of T unless ◦ T ∼ = PSL (
2, p ) with p a Mersenne prime, ◦ T ∼ = PSL (
2, r − ) with r a Fermat prime, ◦ T ∼ = PSL (
2, 8 ) , T ∼ = PSU (
3, 3 ) or T ∼ = PSU (
4, 2 ) .In these cases, S is a Sylow s -subgroup with s = , r , , or , respectively. Artin’sinvestigations were extended by Kimmerle et al. [ ] in 1990 to the cases where T hasexceptional Lie type, with the conclusion that S is always a Sylow p -subgroup. Moreover,as one immediately observes upon inspecting the order formulae for the finite simplegroups of Lie type [ , Table 6], S is always ‘large’ (in both the classical and the exceptionalcases), in the sense that | S | > | T | c for some constant c . Indeed one can take c = by [ , Theorem 3.5]. (On the other hand, | S | | T | by [ , Theorem 3.6].) Date : September 13, 2018.
With the aforementioned question settled, it is natural to ask: how large are the other
Sylow subgroups of T ? Buekenhout [ ] approached this question by investigating the ratios a i = log ( p n ) / log ( p n i i ) in the prime factorisation | T | = Q si = p n i i , where the labelling issuch that p n > p n > · · · > p n s s . Specifically, he asked when a i log ( ) / log ( ) ,calling the corresponding primes p i good contributors to the order of T , and explainingthat the choice of constant log ( ) / log ( ) was in some sense arbitrary but motivated bycomputational evidence and applications in geometry. The conclusion was that if T doesnot have Lie type A , A or A , then | T | has at most one good contributor. Moreprecisely, only such groups have a good contributor r distinct from the characteristic;in all of these cases, r by [ , Theorem 4.1]. The cases where T has type A , A or A produce many further examples, and were left open.In addition to having geometric applications as alluded to in [ ], Buekenhout’s resulthas also recently been used to study certain profinite groups [ ].The purpose of this note is to prove the following result. Theorem . Let q be a prime power, and let T = T ( q ) be a finite simple groupof Lie type, as listed in Table 1 . Let r be a prime dividing | T | but not q , and let R be aSylow r -subgroup of T . Then, for K and M as in Table 1 , we have | R | | T | ( ⌊ log r ( M ) ⌋ + ) /K ,except in the cases listed in the final column of the table. Remark . (i) The conditions listed in the fourth column of Table 1 mitigateoccurrences of isomorphisms between groups in different rows. In particular, note thatG ( ) ∼ = A ( ) .2 , G ( ) ∼ = A ( ) .3 , and that B ( ) is solvable. Note also that weinclude the Tits group F ( ) ′ in Theorem 1.1.(ii) It follows from Table 1 that for all T we have K > ℓ/2 and M ( ℓ + ) , where ℓ is the Lie rank of T as in [ , Tables 5.1.A and 5.1.B]. The upper bound on | R | given inTheorem 1.1 therefore implies the bound | R | | T | ⌊ log r ( ( ℓ + ) r ) ⌋ /ℓ claimed in the abstract.(iii) Theorem 1.1 may be viewed as a refinement of [ , Theorem 4.1], in the followingsense. Let S and R denote the largest and second-largest Sylow subgroups of T , so that | S | = p n and | R | = p n in the above notation. As noted earlier, | S | > | T | c for some constant c . On the other hand, as noted above, Theorem 1.1 implies that | R | | T | c ′ log p2 ( ℓ ) /ℓ forsome constant c ′ , where ℓ is the Lie rank of T . We therefore obtain the following lowerbound on a , which explains why Buekenhout’s good contributors are so rare: a = log | S | log | R | > ℓc log ( | T | ) c ′ log p ( ℓ ) log ( | T | ) = cℓc ′ log p ( ℓ ) . (iv) The questions considered here have also been investigated for the remainingnonabelian finite simple groups, namely the alternating groups and the sporadic simplegroups. Precise answers can, of course, be obtained for the sporadic groups, and arerecorded in [ , Table L.5] and [ , Section 2]. Orders of Sylow subgroups of the alternatinggroup Alt n may be computed using the classical formula of Legendre [ ] for the prime YLOW SUBGROUPS OF FINITE SIMPLE GROUPS 3
Table 1.
Data for the bound | R | | T | ( ⌊ log r ( M ) ⌋ + ) /K in Theorem 1.1, andthe Lie rank ℓ of each group T as per [ , Tables 5.1.A and 5.1.B]. T K M ℓ
Conditions on T ExceptionsA n ( q ) n n + > , ( n, q ) { (
1, 2 ) , (
1, 3 ) } A n ( q ) n/2 2 ( n + ) , 2 | n2n, 2 ∤ n ⌊ n + ⌋ n > , ( n, q ) = (
2, 2 ) B n ( q ) n 2n n n > , ( n, q ) = (
2, 2 ) C n ( q ) n 2n n n > , q oddD n ( q ) n/2 2 ( n − ) n n > D n ( q ) n/2 2n n − > B ( q ) = + , m > D ( q ) ( q, r ) = (
3, 13 ) E ( q )
12 12 6 ( q, r ) = (
3, 13 ) E ( q )
12 18 4 E ( q )
18 18 7 E ( q )
29 30 8 ( q, r ) = (
2, 31 ) F ( q )
12 12 4 ( q, r ) = (
3, 13 ) F ( q ) ′ = + , m > G ( q ) > ( q, r ) = (
3, 13 ) G ( q ) = + , m > factorisation of n !. As one might anticipate, p = and p = almost always, indeedunless n ∈ {
5, 6, 7, 9 } [ , Theorem 3.7]. Moreover, p n ( n ! /2 ) by [ , Table L.4].We now prove some preliminary lemmas in Section 2, before giving the proof ofTheorem 1.1 in Section 3.
2. Supporting lemmas
As in [ , , ], we consider the cyclotomic factorisations for the finite simple groups ofLie type, cf. [ , Definition 4.4]. Writing Φ i for the i th cyclotomic polynomial and d forthe number of diagonal outer automorphisms of T = T ( q ) , this factorisation has the form d | T | = q e M Y i = Φ i ( q ) e i , S. P. GLASBY, ALICE C. NIEMEYER, AND TOMASZ POPIEL
Table 2.
Numbers d of diagonal outer automorphisms, and cyclotomicfactorisations d | T | = q e Q Mi = Φ i ( q ) e i for finite simple classical groups T .The q in Φ i ( q ) are suppressed for brevity. T d = gcd ( · , · ) cyclotomic factorisation of d | T | MA n ( n +
1, q − ) q ( n + ) Φ n1 n + Y i = Φ ⌊ n + ⌋ i n + A n ( n +
1, q + ) q ( n + ) Φ n2 Y = i ≡ ( ) Φ ⌊ ( n + ) i ⌋ i Y i ( ) Φ ⌊ n + lcm ( ) ⌋ i ( n + ) , 2 | n2n, 2 ∤ nB n , C n (
2, q − ) q n Y i = Φ ⌊ lcm ( ) ⌋ i n (
4, q n − ) q n ( n − ) Y i ∤ n and i | Φ − Y i | n or i ∤ Φ ⌊ lcm ( ) ⌋ i ( n − ) D n (
4, q n − ) q n ( n − ) Y i ∤ n Φ ⌊ lcm ( ) ⌋ i Y i | n Φ ⌊ lcm ( ) ⌋ − where e i > for i M , and e M > 0 . We set e i = for i > M . The values of M and e , e , . . . , e M can be deduced from the usual formulae for | T | by noting that q i − = Y k | i Φ k ( q ) and q i + = Y k | ∤ i Φ k ( q ) . They may be also obtained from [ , Definition 4.4 and Tables C.1 and C.2]. The valuesof M are listed in Table 1. Table 2 lists the cyclotomic factorisations for the classicalgroups, and duplicates the values of M for ease of reference.We need the following lemma about cyclotomic polynomials. Lemma . Let i < j be integers, and suppose that r is a prime dividing both Φ i ( q ) and Φ j ( q ) for some prime power q . Then j/i = r k for some positive integer k . Inparticular, r divides j . Proof.
This follows immediately from [ , Theorem 2] or [ , Lemma 2]. (cid:3) The next lemma imposes an upper bound on the number of distinct cyclotomic poly-nomial factors of | T | that can be divisible by a given prime distinct from the characteristic. Lemma . Let T = T ( q ) be a finite simple group of Lie type defined over a field oforder q , and let d | T | = q e Q Mi = Φ i ( q ) e i be the cyclotomic factorisation of T , where d isthe number of diagonal outer automorphisms of T . If r is a prime dividing | T | but not q , YLOW SUBGROUPS OF FINITE SIMPLE GROUPS 5 then r divides at most ⌊ log r ( M/m ) ⌋ + of the factors Φ ( q ) e , . . . , Φ M ( q ) e M , where m is the order of q modulo r . Proof.
Since r divides | T | but not q , it divides some factor Φ i ( q ) e i of d | T | . Hence e i > 0 . Moreover, the minimal such i is the order m of q modulo r . By Lemma 2.1, r might also divide some or all of Φ mr ( q ) e mr , . . . , Φ mr k ( q ) e mrk , where k is maximal suchthat Φ mr k ( q ) divides | T | , but r cannot divide any of the other Φ j ( q ) e j . Since mr k M , r divides at most + k = + ⌊ log r ( M/m ) ⌋ of the factors Φ j ( q ) e j of d | T | . (cid:3) The final lemma bounds the contribution of each cyclotomic polynomial factor to theorder of a finite simple classical group.
Lemma . Let T = T ( q ) be a finite simple classical group defined over a field oforder q , as listed in Table 1 or Table 2 . Let Q ( T ) denote the largest factor of the form Φ i ( q ) e i dividing d | T | . Then Q ( T ) | T | a/n , where a = (cid:14) if T has type A n , B n or C n , if T has type A n , D n or D n . Proof.
Suppose first that n = . Then T = A ( q ) (with q > ) and d | T | = q ( q − ) ,so Q ( T ) = Φ ( q ) . The desired bound is Q ( T ) | T | , and this holds because | T | > q ( q − ) =
12 q ( q − )( q + ) > q + = Φ ( q ) . Suppose from now on that n > . We need the following inequality, which follows from[ , Lemma 3.5]:(1) − q − − q − < ∞ Y i = ( − q − i ) − q − − q − + q − for all q > We now divide the proof into four cases.
Case 1: T ∼ = A n ( q ) with n > . Here d = gcd ( n +
1, q − ) < q (see Table 2) so (1)yields(2) | T | = q n ( n + ) d n + Y i = ( q i − ) = q n ( n + ) d n + Y i = ( − q − i ) > 1 − q − − q − q · q n + . Suppose that a cyclotomic polynomial Φ i ( q ) divides | T | . According to Table 2, we have i n + , e = n and e i = ⌊ ( n + ) /i ⌋ n + for i > . We now showthat Φ i ( q ) e i q n + / ( q − ) . This is true when i = because ( q − ) n + < q n + ; if i n + then Φ i ( q ) divides ( q i − ) / ( q − ) , and so Φ i ( q ) e i (cid:18) q i − − (cid:19) ( n + ) /i < ( q i ) ( n + ) /i ( q − ) ( n + ) /i q n + q − S. P. GLASBY, ALICE C. NIEMEYER, AND TOMASZ POPIEL
Therefore, Q ( T ) q n + / ( q − ) , and it follows from (2) that Q ( T ) n q n ( n + ) ( q − ) n = q n ( n + ) q n ( q − ) n < q n ( n + ) ( q − ) ( − q − − q − ) q n ( n + ) d = | T | . Hence, Q ( T ) | T | as claimed. Case 2: T ∼ = A n ( q ) . Here d = gcd ( n +
1, q + ) q + , so(3) | T | = q n ( n + ) d n + Y i = ( − (− q ) − i ) > q n ( n + ) q + ( + q − ) ∞ Y j = ( − q − ) . That is, for the estimate we have omitted the factors in the original product with odd i > 3 . Using (1), the identity ( + q − ) / ( + q − ) = − q − + q − , and the inequality − q − + q − > , we obtain + q − q + ∞ Y j = ( − q − ) > ( + q − )( − q − − q − ) q ( + q − ) > (cid:18) − q − + q − q (cid:19) > 12q . Together with (3), this yields(4) | T | > 12 q n ( n + )− . We now show that Q ( T ) q ( n + ) . If a cyclotomic polynomial Φ i ( q ) divides d | T | ,then i ( n + ) by Table 2. If i = then e = ⌊ ( n + ) /2 ⌋ ( n + ) /2 (byTable 2), so Φ ( q ) e ( q − ) ( n + ) /2 < q ( n + ) . Similarly, when i = , e = n and Φ ( q ) e < ( q − ) n q ( n + ) . Finally, if i > then e i ( n + ) /i , so Φ i ( q ) e i < ( q i − ) ( n + ) /i < ( q i ) ( n + ) /i = q ( n + ) . Therefore, we have Q ( T ) q ( n + ) , which together with (4) yields Q ( T ) n < q + q + − = (cid:18) q ( n + )− (cid:19) < | T | . Thus, Q ( T ) < | T | n/2 as claimed. Case 3: T ∼ = B n ( q ) or C n ( q ) . Here d = gcd (
2, q − ) , which together with (1) gives | T | = q + n d n Y i = ( − q − ) > 1 − q − − q − · q + n > + n > q + n Suppose that Φ i ( q ) divides d | T | . Then i and e i = ⌊ lcm (
2, i ) ⌋ byTable 2. Since Φ i ( q ) q i − i , we have Φ i ( q ) e i < ( q i ) = q . YLOW SUBGROUPS OF FINITE SIMPLE GROUPS 7
Table 3.
Values of Q ( T ) for the exceptional Lie type groups T , andconstants K , d and q such that Q ( T ) K dd | T | for all q > q . T B D E E E E F F ′ G G Q ( T ) Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ K q Q ( T ) < q , and because n > we have Q ( T ) n < q < q + n | T | . Therefore, Q ( T ) | T | as claimed. Case 4: T ∼ = D n ( q ) or D n ( q ) . Here d = gcd (
4, q ± ) and | T | = q n − n ( q n ± ) d n − Y i = ( q − ) > q − n ∞ Y i = ( − q − ) > ( − q − − q − ) q − n Since Q ∞ i = ( − q − ) > ( − q − − q − ) > by (1), we have | T | > q − n − . If Φ i ( q ) divides d | T | then i and e i by Table 2, so Φ i ( q ) e i ( q i − ) < q .Hence, Q ( T ) q , and so Q ( T ) n q < ( q − n − ) < | T | . Whence, Q ( T ) | T | as claimed. This completes the proof. (cid:3)
3. Proof of Theorem 1.1
Proof of Theorem 1.1.
As in Section 2, write d | T | = q e Q Mi = Φ i ( q ) e i and let Q ( T ) denote the maximum of Φ ( q ) e , . . . , Φ M ( q ) e M . Let r be a prime dividing | T | butnot q , and let R be a Sylow r -subgroup of T . It follows from Lemma 2.2 that(5) | R | Q ( T ) + ⌊ log r ( M ) ⌋ . We now seek ‘large’ constants K satisfying Q ( T ) K | T | , in order to deduce the claimedbounds of the form | R | | T | ( ⌊ log r ( M ) ⌋ + ) /K .Suppose first that T is a classical group. Then the values of M may be deduced fromthe cyclotomic factorisations listed in Table 2, and are as listed in both Table 1 andTable 2. The values of K appearing in Table 1 are obtained from Lemma 2.3.Now suppose that T is an exceptional Lie type group. Then d | T | = q e Q = Φ i ( q ) e i ,where the values of e , . . . , e are listed in [ , Table C.2]; in particular, we obtain thevalues of M appearing in Table 1. The values of e appear in the last row of [ , Table C.3], S. P. GLASBY, ALICE C. NIEMEYER, AND TOMASZ POPIEL and in [ , Table 6]. By inspecting these factorisations, one finds the value of i such that Q ( T ) = Φ i ( q ) e i (and that this value is independent of q ). Table 3 lists constants K , d and q such that Φ i ( q ) e i K dd | T | for all q > q . Note that d = d = in all cases, except when T = E ( q ) , E ( q ) or E ( q ) , where ( d, d ) = ( gcd (
3, q − ) , 3 ) , ( gcd (
3, q + ) , 3 ) or ( gcd (
2, q − ) , 2 ) , respectively. Inparticular, d d in all cases, and so Q ( T ) K | T | for all q > q . The constants K agree with those in Table 1, and so by combining the above boundwith (5) we obtain the claimed bounds | R | | T | ( ⌊ log r ( M ) ⌋ + ) /K for q > q . It remains toconsider the cases where q < q . In these cases, we check manually, for each prime r dividing | T | but not q , whether the bound | R | | T | ( ⌊ log r ( M ) ⌋ + ) /K (with K and M as inTable 1) holds. The exceptions, which were checked both manually and using the Magma code available at the first author’s website , are recorded in Table 1. (cid:3) Remark . One can slightly improve the values of K listed in Table 1 in some cases.For example, if T ∼ = B ( q ) then K can be increased to log ( | B ( ) | ) / log ( Φ ( )) ≈ .As such improvements seem tedious to achieve and do not change the form of our genericbound | R | | T | O ( log r ( ℓ ) /ℓ ) , where ℓ is the Lie rank of T , we chose not to pursue them here. Acknowledgements
All three authors acknowledge support from the Australian Research Council (ARC)grant DP140100416, and SPG also acknowledges support from DP160102323. SPG andTP are grateful to RWTH Aachen University for financial support and hospitality duringtheir respective visits in 2017, when the research leading to this paper was undertaken.
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Centre for Mathematics of Symmetry and Computation, The Universityof Western Australia, 35 Stirling Highway, Perth WA 6009, Australia. Also affiliatedwith The Department of Mathematics, University of Canberra, ACT 2601, Australia.Email: ∼ glasby/ (A. C. Niemeyer) Lehrstuhl B f¨ur Mathematik, Lehr- und Forschungsgebiet AlgebraRWTH Aachen University, Pontdriesch 10-16, 52062 Aachen, Germany.Email: (T. Popiel)
School of Mathematical Sciences, Queen Mary University of London, MileEnd Road, London E1 4NS, United Kingdom. Also affiliated with the Centre for Mathe-matics of Symmetry and Computation, The University of Western Australia, 35 StirlingHighway, Crawley WA 6009, Australia. Email: [email protected]@uwa.edu.au