the primitive permutation groups of squarefree degree

Bull. London Math. Soc. 35 (2003) 635–644 C2003 London Mathematical SocietyDOI: 10.1112/S0024609303002145

THE PRIMITIVE PERMUTATION GROUPS OF SQUAREFREEDEGREE

CAI HENG LI and AKOS SERESS

Abstract

The paper gives lists of all the primitive permutation groups of squarefree degree. All such groups areeither solvable and act on a prime number of points, or are almost simple. Among the almost simpleexamples, the groups of Lie type have rank at most 2, or the point stabilizer is a parabolic subgroup.

1. Introduction

We determine all the primitive permutation groups G Sn with n a squarefreenumber. Similar lists already exist for primitive groups of odd degree [16], andfor groups where n is divisible by a prime greater than

√n (see [15]). These lists

have proved to be useful in applications; our primary motivation for compiling thesquarefree list is the determination of all non-Cayley graphs of squarefree order thathave primitive automorphism groups [13]. That investigation is part of a programinitiated by Marusic [21], which aims to determine all the integers n that are theorders of vertex-transitive, non-Cayley graphs. The status of all non-squarefree n

has already been settled [22]. Another possible application is the investigation ofedge-transitive graphs with a squarefree number of vertices.

Theorem 1. Let G Sn be primitive and of squarefree degree. Then either

(i) n is a prime and Zn G AGL(1, n), or(ii) G is almost simple and Soc(G) appears in one of Tables 1–4.

We note that although Tables 3 and 4 include all the almost simple primitivegroups of squarefree degree that have Lie-type socle, the parameters q, m, and k inthese tables must satisfy further arithmetic conditions in order for the degree to besquarefree. For each line of these two tables, there are values of the parameters suchthat the primitive groups described there have squarefree degree.

The primitive groups of squarefree degree n, where n is the product of at mostthree distinct primes, have already been determined, and are given in [15], [25],and [4].

The O’Nan–Scott theorem [18, 24] classifies primitive permutation groups G Sninto the following five categories:

(i) n = pd and Zdp G AGL(d, p);

(ii) G is almost simple;

Received 7 June 2002; revised 9 October 2002.

2000 Mathematics Subject Classification 20B15.

The first author’s research is partially supported by an ARC Fellowship and an ARC Large Grant.The second author’s research is partially supported by the NSF.

636 cai heng li and akos seress

Table 1. Groups with alternating socle.

Soc(G) n Action Comment

1 Ac

(ck

)on k-sets n squarefree

2 A8 105 on Π(2, 4) Π(a, b) := set of partitions

of [1, ab] into b sets of size a

3 A6 15 on Π(2, 3)

4 A2a12

(2aa

)on Π(a, 2) a ∈ 3, 4, 6, 9, 10, 12, 36

5 A8 15 Gα ∩ Soc(G) = AGL(3, 2)

6 A7 15 Gα ∩ Soc(G) = GL(3, 2)

7 A5 6 Gα ∩ Soc(G) = D10

(iii) G has a regular normal subgroup Tk for some nonabelian simple group T

and k 1;(iv) Soc(G) = Tk for some nonabelian simple group T , k 2, and n = |T |k−1;(v) G A St for some group A of type (ii) or (iv) and t > 1, and n = |A : H |t

for some subgroup H A.

It is clear that groups in (iii)–(v) cannot have squarefree degree, and the squarefreeexamples in (i) are the ones described in Theorem 1(i). We determine the groupsof squarefree degree belonging to case (ii) in the next four sections. The necessarygroup-theoretical information needed for this investigation is mostly contained inthe literature; in most cases, our task is to estimate subgroup orders and checkarithmetic conditions.

For the rest of the paper, let G Sym(Ω) be almost simple, acting primitivelyon Ω. Let n denote |Ω|. For α ∈ Ω, the stabilizer Gα is a maximal subgroup of G; weare concerned about the possibilities for Gα ∩ Soc(G), provided that n is squarefree.

2. The case of alternating socle

Suppose that Soc(G) = Ac for some c 5. If c = 6, then we temporarily assumethat G S6. We distinguish three cases, according to the behavior of Gα in thenatural action on c points.

If Gα is intransitive in the natural action of Sc on c points, then Gα ∩ Soc(G) =(Sk × Sc−k) ∩ Ac for some k with 1 k < c, n = (c

k), and we are in the case described

on line 1 of Table 1. Squarefree binomial coefficients have been extensively studied;see [6] and its references. Here we mention only that if (c

k) is squarefree, then k/c → 0

as c → ∞, but there are infinitely many values c such that (ck) is squarefree for all

k ln n/5.If Gα is transitive, but imprimitive in the natural action on c points, then c = ab

for some integers a, b with 2 a, b c/2, such that Gα stabilizes a partition of [1, c]into b sets of size a. In this case, n is the number of partitions of [1, ab] into b setsof size a; that is,

n =(ab)!

(a!)bb!. (1)

If both a and b are greater than or equal to 4, then n is not squarefree. Indeed, byChebyshev’s theorem, there exists a prime p such that ab/4 < p < ab/2, and so p2|n.If a = 3 and b 4, then 52|n, and checking the values b = 2, 3 indicates that the

the primitive permutation groups of squarefree degree 637

only example with a = 3 is (a, b) = (3, 2), included on line 4 of Table 1. If a = 2 andb 5, then 32|n and checking the remaining small values of b leads to lines 2 and 3of Table 1. The case where b = 3 can be handled by the following strengthening ofChebyshev’s theorem.

Lemma 2. For n 8, there is a prime p satisfying n < p < 3n/2.

Proof. For n > 3.06 · 108, this follows from [2, Theorem 3.4], if we take theparameter value k = n/4 in that theorem. For 8 n 3.06 ·108, one of the primes11, 13, 19, 23, 31, 43, 61, 89, 131, 193, 283, 421, 631, 941, 1409, 2113, 3169, 4751,7121, 10667, 15991, 23981, 35969, 53951, 80923, 121379, 182059, 273083, 409609,614413, 921611, 1382393, 2073563, 3110339, 4665499, 6998221, 10497299, 15745943,23618891, 35428333, 53142491, 79713707, 119570551, 179355823, 269033731 and403550593 is between n and 3n/2, since this sequence of primes satisfies the require-ment that the ratio of consecutive members be less than 3/2.

Lemma 2 implies that if b = 3 and a 8, then the number n in (1) is notsquarefree, since for a prime p satisfying a < p < 3a/2 we have p2|n. The values2 a 7 give only the example on line 3 of Table 1. If b = 2, then 22|n unless thebinary decomposition of a contains at most two 1’s. This follows from the fact thatthe exponent of 2 in (2a

a)/2 is

∑k1

(⌊2a

2k

⌋− 2

⌊a

2k

⌋)− 1,

and the kth term of this sum is 1 if the kth digit from the left of the binary expansionof a is 1, and 0 otherwise. Moreover, by [6, Theorem 1∗], if a 2082 then (2a

a)/2 is

divisible by the square of a prime greater than√a/5. Checking the values a = 2i

and a = 2i + 2j for a < 2082 in GAP [5] gives the examples on line 4 of Table 1.The third possibility for Gα is that Gα acts primitively in the natural representation

of Sc on c points. In this case, |Gα| < 4c by [23]. Moreover, using the fact that theproduct of primes at most c is less than 4c, and the fact that the squarefree numbern = |G : Gα| is the product of such primes, we obtain n < 4c. Combining the twoinequalities, we get |G| < 16c. On the other hand, |G| c!/2 > (c/3)c; summarizing,we obtain c/3 < 16. The primitive groups of degree c < 48 are known explicitly,and can be found for example in the GAP library. Checking this list, we see that theonly examples are the ones listed in lines 5, 6, and 7 of Table 1.

Finally, in the case c = 6, we check in the Atlas [3] that the maximal subgroupsof groups with Soc(G) = A6, G S6, add no further examples to Table 1.

3. The case of sporadic socle

Suppose that Soc(G) is a sporadic simple group. The list of maximal subgroups isknown for all simple sporadic groups except the Monster, so it is trivial to find themaximal subgroups of G of squarefree index. In most cases, the lists are containedin the Atlas [3]; the ambiguities in the cases J4, Th, Fi22, Fi23, Fi24, and B noted in[3] are resolved in [11], [19], [10], [12], [20], and [27], respectively. Moreover, the listof local subgroups of the Monster is known (see [3] and work on local 2-subgroups,currently in preparation by Meierfrankenfeld and Shpectorov). Therefore, our onlytask is to prove that the Monster has no nonlocal subgroups of squarefree index.


Table 2. Groups with sporadic socle.

Soc(G) n Gα ∩ Soc(G)

1 M11 11 M10

2 M11 55 M9 : 2

3 M11 165 M8 : S3

4 M12 66 M10 : 2

5 M22 22 PSL(3, 4)

6 M22 77 24 : A6

7 M22 231 24 : S5

8 M22 330 23 : PSL(3, 2)

9 M23 23 M22

10 M23 253 PSL(3, 4) : 22 or 24 : A7

11 M23 506 A8

12 M23 1771 24 : (3 × A5) : 2

13 M24 759 24 : A8

14 M24 1771 26 : 3.S6

15 M24 3795 26 : (PSL(3, 2) × S3)

16 J1 266 PSL(2, 11)

17 J1 1045 23 : 7 : 3

18 J1 1463 2 × A5

19 J1 2926 D6 × D10

Lemma 3. Let H be a nonlocal subgroup of the Monster M, with Soc(H) =T1 × . . .×Tr for some nonabelian simple groups Ti. Then one of the following occurs:

(i) r = 1;(ii) Soc(H) = T × A5 with T ∈ PSL(2, 19),PSU(3, 8),HN;(iii) Soc(H) = T × PSL(3, 2) with T ∈ PSL(2, 16),PSL(2, 17),PSp(4, 4),He;(iv) Soc(H) = PSL(3, 3) × PSL(3, 3);(v) Soc(H) = T1 × T2 with Ti ∈ M11,M12,PSL(2, 11);(vi) Soc(H) = T1 × T2 with

T1 ∈ A11, A12,PSL(2, 11),PSU(5, 2),M11,M12,M22,HS,McL andT2 ∈ A5, A6;

(vii) 2 r 8, and there is no prime p > 7 dividing the order of any of the |Ti|.

Proof. Let p be the largest prime dividing | Soc(H)|, and let g ∈ Soc(H) with|g| = p. Without loss of generality, we can suppose that g ∈ T1. Then CM(g) 〈g〉×T2×. . .×Tr . Checking the centralizer orders in [3], we see that |CM(g)|/|〈g〉| 24if p 23. Therefore, if p 23, then r = 1.

If p = 19, then |CM(g)|/|〈g〉| = 60. Hence r = 1, or r = 2 and T2 = A5. Inthe latter case, the order of the centralizer of an element h ∈ T2 with |h| = 5is divisible by 19, implying that h is in the conjugacy class 5A, and |T1| divides|CM(h)|/|〈h〉| = 214 · 36 · 56 · 7 · 11 · 19. The list of simple groups involved in theMonster is also given in the Atlas; checking this list, we obtain the possibilitiesdescribed in case (ii).

Similarly, if p = 17 then |CM(g)|/|〈g〉| = 168, implying that r = 1, or r = 2 andT2 = PSL(3, 2). In the latter case, the order of the centralizer of an element h ∈ T2

with |h| = 7 is divisible by 17, implying that h is in the conjugacy class 7A, and |T1|


divides |CM(h)|/|〈h〉| = 210 · 33 · 52 · 74 · 17. This leads to the possibilities described incase (iii).

If p = 13, then we have |CM(g)|/|〈g〉| = | PSL(3, 3)|. Noting that the onlynonabelian simple group whose order divides | PSL(3, 3)| is PSL(3, 3), we find that ifr > 1, then T2 = PSL(3, 3). Since T2 also contains an element of order 13, the sameargument shows that T1 = PSL(3, 3), and we are in case (iv).

If p = 11, then |CM(g)|/|〈g〉| = |M12|. The nonabelian simple groups whose orderdivides |M12| are A5, A6, M11, M12, and PSL(2, 11), and the product of the orders ofany two of these groups does not divide |M12|. Hence r 2 in this case as well. Ifr = 2 and 11

∣∣|T2|, then the same argument shows that T1 ∈ M11,M12,PSL(2, 11),and we are in case (v). If 11 | |T2| then T2 ∈ A5, A6. The order of the centralizerof any h ∈ T2 with |h| = 5 is divisible by 11, so h is in the conjugacy class 5A and|T1| divides 214 · 36 · 56 · 7 · 11. Checking again the list of simple groups involved inthe Monster, we obtain the possibilities described in (vi).

Finally, if p 7, then examining the orders of the centralizers of elements in theconjugacy classes 5A, 5B, 7A and 7B reveals that the 2-part of |CM(g)|/|〈g〉| is atmost 214, implying that r − 1 14/2 = 7.

Lemma 3 implies immediately that M has no nonlocal subgroup H of squarefreeindex. If H belongs to cases (i)–(vi) then 22

∣∣|M : H |, and if H belongs to case (vii)then 132

∣∣|M : H |.

4. The case of classical socle

Let G be almost simple, with L = Soc(G) a classical group of Lie type ofcharacteristic p. Aschbacher [1] defined a category C(G) of geometric subgroupsof G and a category S(G) of almost simple subgroups of G satisfying certainirreducibility conditions in the action on the natural module of definition of G. It isshown in [1] that each maximal subgroup of G belongs to either C(G) or S(G). Inthe construction of Table 3, our main tool is the following theorem of Liebeck [14].

Theorem 4. Let L be a classical simple group, with associated geometry ofdimension d over the field of definition . Let G satisfy L G Aut(L), and letH be a maximal subgroup of G. Then one of the following holds:

(i) H ∈ C(G);(ii) H ∈ S(G) is Ak or Sk , with d + 1 k d + 2;(iii) H ∈ S(G) and |H | < ||3d.

It turns out that if a maximal subgroup H belongs to cases (i) or (ii) thenp2

∣∣|G : H |, unless H is a parabolic subgroup, or (G,H) is one of the low-rankexamples occurring on lines 3–8 or 12 of Table 3. Most of the parabolic subgroupsare also eliminated because the index |G : H | is divisible by the square of acyclotomic polynomial value Φl(||). Finally, a simple estimate on the orders of Hwith squarefree index shows that there are no examples in case (iii). All of thesecomputations are straightforward, but quite tedious. We give details only in the caseSoc(G) = PSp(2m, q), m 2; the other types of socles can be handled analogously.

The category C(G) is divided into eight subcategories C1–C8. We shall use thesubdivision from [9], where it is also worked out which members of C(G) are actuallymaximal subgroups in G. For the symplectic case, the results are in [9, Table 3.5.C].


Table 3. Groups with classical socle.

Soc(G) n Action Comment

1 PSL(m, q)∏k−1

i=0 (qm−i−1)∏ki=1(qi−1)

on k-dim. subspaces 1 k < m

2 PSL(m, q)∏2k−1

i=0 (qm−i−1)(∏ki=1(qi−1)

)2 on (k, m − k)-dim. flags 1 k < m/2

3 PSL(2, q) q(q + 1)/2 Gα ∩ Soc(G) = D2(q−1)/d d = (2, q − 1)

4 PSL(2, q) q(q − 1)/2 Gα ∩ Soc(G) = D2(q+1)/d d = (2, q − 1)

5 PSL(2, q) q0

(q2

0 + 1)/2 Gα ∩ Soc(G) = PGL(q0) q = q2

0 odd

6 PSL(2, q) q(q2 − 1)/24 Gα ∩ Soc(G) = A4 q ≡ ±3 (8)

7 PSL(2, q) q(q2 − 1)/48 Gα ∩ Soc(G) = S4 q ≡ ±1 (8)

8 PSL(2, q) q(q2 − 1)/120 Gα ∩ Soc(G) = A5 q ≡ ±1 (10)

9 PSU(m, q) (qm−(−1)m)(qm−1−(−1)m−1)q2−1

on t.i. 1-spaces

10 PSp(2m, q) q2m−1q−1 on (t.i.) 1-spaces

11 PSp(2m, q) (q2m−1)(q2m−2−1)(q2−1)(q−1)

on t.i. 2-spaces

12 PSp(4, 2)′ 10 on V2 ⊥ V2, ⊥ denotes

V2 nonsing. orthogonal

on cosets of Ω+(4, 2) decomposition

13 PSp(4, 2)′ 6 on cosets of Ω−(4, 2)

14 Ω(2m + 1, q) q2m−1q−1 on t.s. 1-spaces

15 Ω(2m + 1, q) (q2m−1)(q2m−2−1)(q2−1)(q−1)

on t.s. 2-spaces

16 PΩ−(2m, q) (qm+1)(qm−1−1)q−1 on t.s. 1-spaces m even

17 PΩ−(2m, q) (qm+1)(q2m−2−1)(qm−2−1)(q2−1)(q−1)

on t.s. 2-spaces 2|q

18 PΩ−(2m, q) (qm+1)(q2m−2−1)(q2m−4−1)(qm−3−1)(q3−1)(q2−1)(q−1)

on t.s. 3-spaces 2|q, 4|m

19 PΩ+(2m, q) (qm−1)(qm−1+1)q−1 on t.s. 1-spaces m odd

20 PΩ+(2m, q) (qm−1)(q2m−2−1)(qm−2+1)(q2−1)(q−1)

on t.s. 2-spaces 2|q

21 PΩ+(2m, q) (qm−1)(q2m−2−1)(q2m−4−1)(qm−3+1)(q3−1)(q2−1)(q−1)

on t.s. 3-spaces 2|q, m ≡ 3 (4)

For notational convenience, in most of the nonparabolic cases we write a maximalsubgroup H in the central extension of G. Since we estimate the exponent of thecharacteristic prime p in |G : H |, this extension does not influence the validity of theargument.

Category C1: stabilizers of totally isotropic or nonsingular subspaces. If H is thestabilizer of a k-dimensional totally isotropic (t.i.) subspace, then |G :H | is thenumber of such subspaces, which is

∏k−1i=0 (q2m−2i − 1)∏k

i=1(qi − 1)

=

∏k−1i=0 (qm−i − 1)∏ki=1(q

i − 1)

k−1∏i=0

(qm−i + 1).


The fraction on the right-hand side of this identity is an integer. If k 4, or k= 3and m is odd, then at least two of the numbers m − i for 0 i k − 1 areodd, and (q + 1)2 divides

∏k−1i=0 (qm−i + 1). If k = 3 and m is even, then (q2 + 1)2

divides the numerator on the left-hand side, so the square of any odd prime r

dividing q2 + 1 divides |G : H |. For k = 1, 2, it is possible that the number of totallyisotropic k-spaces is squarefree, and these possibilities are listed on lines 10 and 11 ofTable 3.

If H is the stabilizer of a nonsingular subspace, then H = Sp(2k, q) ⊥ Sp(2m−2k, q)for some k < m/2. The exponent of q in |H | is k2 + (m− k)2 m2 − 2, so q2

∣∣|G : H |.

Category C2: stabilizers of decompositions into subspaces of identical size. Onepossibility is that m = kt with t 2, and H stabilizes a decomposition into t

perpendicular subspaces of dimension 2k. Then Soc(G) ∩ H = Sp(2k, q) St, and theexponent of q in | Soc(G) ∩ H | is less than tk2 + t because t! is not divisible by thetth power of any prime. Moreover, if m = kt 3, then m2 − (tk2 + t) 2, and soq2

∣∣|G : H |. If m = 2, then the p-part of | Soc(G) ∩ H | is q2 times the p-part of 2!,which is less than the p-part of q4 by at least a factor p2 unless p = q = 2. Thisexception leads to line 12 of Table 3; note that this example has already occurredon line 4 of Table 1.

The other possibility is that H stabilizes a decomposition into two m-dimensionaltotally isotropic subspaces, Soc(G)∩H = GL(m, q).2, and q is odd. Then the exponentof q in |G : H | is m2 − m(m − 1)/2 2.

Category C3: subgroups defined over field extensions. One possibility is that m = kt

for some prime t, and Soc(G) ∩ H = Sp(2k, qt). Then the exponent of q in |G : H | ism2 − tk2 = (t − 1)tk2 2. The other possibility is that q is odd, and Soc(G) ∩ H =GU(m, q). Then the exponent of q in |G : H | is m2 − m(m − 1)/2 2.

Category C4: tensor product subgroups. Then q is odd, then m = kt with t 3,and Soc(G) ∩ H = Sp(2k, q) ⊗ Oε(t, q). The exponent of q in |Oε(t, q)| is (t2 − 2t)/4if t is even, and (t2 − 2t + 1)/4 if t is odd. Hence, in any case, the exponent of q in|G : H | is at least (kt)2 − k2 − (t − 1)2/4 = (t − 1)((t + 1)k2 − (t − 1)/4) 2.

Category C5: subgroups defined over subfields. Then Soc(G) ∩ H = Sp(2m, q0),where q = qt0 with t prime. The exponent of q0 in |G : H | is (t − 1)m2 2.

Category C6: symplectic-type normalizers. In this case, q = p is odd, 2m = 2k , andSoc(G) ∩ H = 21+2kO−(2k, 2). The odd part of |O−(2k, 2)| is (2k + 1)

∏k−1i=1 (22i − 1) <

22k2

. We have qm2−2 = q22k−2−2 322k−2−2. Therefore, if 322k−2−2 > 22k2

, then |G : H |is divisible by q2. This inequality holds for k 3, since for k 4 the exponent22k−2 − 2 is already greater than 2k2, and for k = 3 we have 314 > 218. If k = 2,then |O−(2k, 2)| = 60, which is divisible by at most the first power of q, and soq3

∣∣|G : H |.

Category C7: tensor power subgroups. Now 2m = (2k)t with t 3, tq is odd,and Soc(G) ∩ H = PSp(2k, q) St. The exponent of q in |G : H | is greater thanm2 − tk2 − t = 22t−2k2t − tk2 − t 22t−2 − 2t 2.

Category C8: classical subgroups. In this case, q = 2e for some e 1; moreover,Soc(G)∩H = Oε(2m, q) if (m, e) = (2, 1), and Soc(G)∩H = Ωε(2m, q) if (m, e) = (2, 1).If (m, e) = (2, 1), then the exponent of 2 in |G : H | is e(m2−m(m−1))−1 = em−1 > 1.


Table 4. Groups with exceptional socle.

Soc(G) n Gα ∩ Soc(G) Comment

1 2B2(q) q2 + 1 q2.(q − 1)

2 G2(q) (q + 1)(q4 + q2 + 1) q5. SL(2, q).(q − 1)

3 3D4(q) (q + 1)(q8 + q4 + 1) q9. SL(2, q3).(q − 1)

4 F4(q)(q12−1)(q4+1)

q−1 q9. Sp(6, q).(q − 1) q = 4e

5 2E6(q)(q12−1)(q4+1)(q6−q3+1)

q−1 q21.(2, q + 1).PSU(6, q).(q − 1) q = 4e

6 E7(q)(q18−1)(q14−1)(q4−q2+1)

(q2−1)(q−1)q33.(O+(12, q).(q − 1))/(2, q − 1)

If Soc(G) = Sp(4, 2)′ ∼= A6, then | Soc(G) : Ω+(4, 2)| = 10 and | Soc(G) : Ω−(4, 2)| = 6,and these possibilities are recorded on lines 12 and 13 of Table 3.

Next, we prove that if H belongs to case (ii) of Theorem 4, then |G : H | is notsquarefree. The exponent of q in (2m + 2)! is at most 2m + 1, so we have done ifm2 −2 2m+1. This inequality holds for m 3. For m = 2, we have PSp(4, 2)′ = A6.If m = 2 and q > 2, then | Soc(G) : A6| is divisible by at least the third power of aprime.

Finally, we show that there are no maximal subgroups of squarefree index incase (iii). The order of any simple group L of Lie type can be written in the form

|L| =1

dqh

∏k

Φk(q)rk , (2)

where q is the size of the field of definition (the square root of the size in the caseof unitary groups), h, k and rk are positive integers, Φk(x) is the kth cyclotomicpolynomial, and d is a divisor of q − 1 or q + 1. In particular, if Soc(G) = L and|G : H | is squarefree, then

|L ∩ H | 1

dqh−1

∏k

Φk(q)rk−1. (3)

In the case L = PSp(2m, q), estimate (3) implies that if m 7, then |L∩H | qm2−1 >

q3·2m, and so H cannot belong to case (iii). For m = 5 and m = 6, a refinement ofthis estimate using the other terms in the right-hand side of estimate (3) proves that|L ∩ H | > q3·2m. For 2 m 4, the maximal subgroups of PSp(2m, q) are listed inKleidman’s thesis [7], and it is easy to check that no maximal subgroup in S(G)has squarefree index. This finishes the proof of correctness of Table 3 in the case ofsymplectic socles.

5. The case of exceptional socle

Let G be almost simple, with L(q) = Soc(G) an exceptional group of Lie typeof characteristic p, defined over GF(q). In the rank 1 cases L(q) = 2B2(q) andL(q) = 2G2(q), we check the list of maximal subgroups in [26] and [8], respectively;the only example with squarefree index is listed on line 1 of Table 4. For exceptionalsocles of rank at least 2, we appeal to a result of Liebeck and Saxl [17]. This resultasserts that if H is a maximal subgroup of G with |H | qk(L(q))|G : L(q)|, then H


is parabolic, or H appears in a list given in [17]. The number k(L(q)) exceeds theexponent of q in the order formula (2) by at most 1, so the estimate (3) implies thatevery maximal subgroup H of squarefree index must satisfy |H | qk(L(q))|G : L(q)|.All subgroups in the list in [17] have index divisible by q2. Parabolic subgroupswhose index may be squarefree are listed in lines 2–6 of Table 4.

We give details of this argument only in the case L(q) = 2E6(q), with q = pe. Inthis case, k(2E6(q)) = 37, |G : L(q)| 2e(3, q + 1), and the estimate (3) implies thatevery maximal subgroup H of squarefree index must satisfy

|H | q35Φ1(q)3Φ2(q)

5Φ3(q)Φ4(q)Φ6(q)2/(3, q + 1).

This bound is obviously greater than 2e(3, q + 1)q37.The maximal nonparabolic subgroups of order greater than 2e(3, q + 1)q37 are

F4(q), (O−(10, q) (q+1)/(3, q+1)).(4, q+1), (SL(2, q)SU(6, q)).(2, q−1), and, in the

case q = 2, Fi22. In each of these cases, |G : H | is divisible by q12. The possibilitiesH ∩ Soc(G) for maximal parabolic subgroups H are

H ∩ Soc(G) = q29.(SL(3, q2)/(3, q + 1) × SL(2, q)).(q − 1),

H ∩ Soc(G) = q31.(SL(3, q) × SL(2, q2)).(q2 − 1)/(3, q + 1),

H ∩ Soc(G) = q24.O−(8, q).(q2 − 1)/(3, q + 1),

and

H ∩ Soc(G) = q21.(2, q + 1).PSU(6, q).(q − 1).

In the first three of these cases, Φ6(q)2 divides |G : H |, whereas in the fourth case

|G : H | = Φ2(q)Φ3(q)Φ4(q)Φ6(q)Φ8(q)Φ12(q)Φ18(q).

If q is odd, then 22|Φ4(q)Φ8(q), and if q is a power of two with odd exponent, then32|Φ2(q)Φ6(q). Finally, if q is a power of two with even exponent, then |G : H | maybe squarefree; this possibility is recorded in line 5 of Table 4.

References

1. M. Aschbacher, ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984)469–514.

2. A. Balog, C. Bessenrodt, J. B. Olsson and K. Ono, ‘Prime power degree representations of thesymmetric and alternating groups’, J. London Math. Soc. (2) 64 (2001) 344–356.

3. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of finite groups(Oxford University Press, Eynsham, 1985).

4. G. Gamble and C. E. Praeger, ‘Vertex-primitive groups and graphs of order twice the product oftwo distinct odd primes’, J. Group Theory 3 (2000) 247–269.

5. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3 (The GAP Group,Aachen/St Andrews, 2002); http://www-gap.dcs.st-and.ac.uk/~gap.

6. A. Granville and O. Ramare, ‘Explicit bounds on exponential sums and the scarcity of squarefreebinomial coefficients’, Mathematika 43 (1996) 73–107.

7. P. Kleidman, ‘The subgroup structure of some finite simple groups’, PhD thesis, Cambridge University,1987.

8. P. Kleidman, ‘The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups2G2(q), and their automorphism groups’, J. Algebra 117 (1988) 30–71.

9. P. Kleidman and M. Liebeck, The subgroup structure of the finite classical groups (CambridgeUniversity Press, Cambridge, 1990).

10. P. B. Kleidman and R. A. Wilson, Corrigendum: ‘The maximal subgroups of Fi22’, Math. Proc.Cambridge Philos. Soc. 102 (1987) 17–23.

11. P. B. Kleidman and R. A. Wilson, ‘The maximal subgroups of J4’, Proc. London Math. Soc. (3) 56(1988) 484–510.


12. P. B. Kleidman, R. A. Parker and R. A. Wilson, ‘The maximal subgroups of the Fischer groupFi23’, J. London Math. Soc. (2) 39 (1989) 89–101.

13. C. H. Li and A. Seress, ‘On vertex-primitive non-Cayley graphs of squarefree order’, 2002, submitted.14. M. W. Liebeck, ‘On the orders of maximal subgroups of the finite classical groups’, Proc. London

Math. Soc. (3) 50 (1985) 426–446.15. M. W. Liebeck and J. Saxl, ‘Primitive permutation groups containing an element of large prime

order’, J. London Math. Soc. (2) 31 (1985) 237–249.16. M. W. Liebeck and J. Saxl, ‘The primitive permutation groups of odd degree’, J. London Math. Soc.

(2) 31 (1985) 250–264.17. M. W. Liebeck and J. Saxl, ‘On the orders of maximal subgroups of the finite exceptional groups of

Lie type’, Proc. London Math. Soc. (3) 55 (1987) 299–330.18. M. W. Liebeck, C. E. Praeger and J. Saxl, ‘On the O’Nan–Scott theorem for finite primitive

permutation groups’, J. Austral. Math. Soc. Ser. A 44 (1988) 389–396.19. S. A. Linton, ‘The maximal subgroups of the Thompson group’, J. London Math. Soc. (2) 39 (1989)

79–88.20. S. A. Linton and R. A. Wilson, ‘The maximal subgroups of the Fischer groups Fi24 and Fi′24’, Proc.

London Math. Soc. (3) 63 (1991) 113–164.21. D. Marusic, ‘Cayley properties of vertex symmetric graphs’, Ars Combin. (B) 16 (1983) 297–302.22. B. D. McKay and C. E. Praeger, ‘Vertex-transitive graphs that are not Cayley graphs. II’, J. Graph

Theory 22 (1996) 321–334.23. C. E. Praeger and J. Saxl, ‘On the orders of primitive permutation groups’, Bull. London Math. Soc.

12 (1980) 303–307.24. L. L. Scott, ‘Representations in characteristic p’, The Santa Cruz Conference on Finite Groups, Proc.

Sympos. Pure Math. 37 (Amer. Math. Soc., Providence, RI, 1980) 319–331.25. A. Seress, ‘On vertex-transitive, non-Cayley graphs of order pqr’, Discrete Math. 182 (1998) 279–292.26. M. Suzuki, ‘On a class of doubly transitive groups’, Ann. Math. 75 (1962) 105–145.27. R. A. Wilson, ‘The maximal subgroups of the Baby Monster. I’ J. Algebra 211 (1999) 1–14.

Cai Heng LiUniversity of Western AustraliaCrawley, WA 6009Australia

[email protected]

Akos SeressThe Ohio State UniversityColumbus, OH 43210USA

[email protected]

the primitive permutation groups of squarefree degree

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