JOURNAL OF ALGEBRA 188, 516᎐530Ž. 1997 ARTICLE NO. JA966828

Ž.p,q,r-Generations of the Smallest Conway

Group Co3

Shahiem Ganief*†

Uni¨ersity of the Western Cape, Pri¨ate Bag X17, Bell¨ille, 7535, South Africa View metadata, citation and similar papers at core.ac.uk brought to you by CORE

and provided by Elsevier - Publisher Connector

Jamshid Moori*‡

Uni¨ersity of Natall Pri¨ate Bag X01, Scotts¨ille, Pietermaritzburg, 3209, South Africa

Communicated by Walter Feit

Received October 15, 1995

An Ž.l, m, n -generated group G is a quotient group of the triangle group Tl Ž,m, lmn n.²sx,y,zNxsyszsxyz s 1: . In this paper the authors continue the study on the Ž.p, q, r -generations of the sporadic simple groups, where p, q, r are

distinct primes. The problem is resolved for the groups Co3. ᮊ 1997 Academic Press

1. INTRODUCTION

A group G is said to be Ž.l, m, n -generated if G s ²:Ž.x, y , with oxsl, oyŽ.sm, and oxy Ž .sn. In this case G is a quotient group of the triangular group TlŽ.,m,n, and by the definition of the triangular group, ŽŽ.Ž.Ž.. Gis also ␲ l , ␲ m , ␲ n -generated for any ␲ g S3. We may therefore assume l F m F n. It is also well known that if G is an Ž.l, m, n -generated simple group, then 1rl q 1rm q 1rn - 1Žcf.1.wx. In a series of papers the authors established theŽ. 2, 3, p --generations of

the sporadic simple group Fcf22 Ž .wx 11.Ž. and the p, q, r -generations of the

* The first author was supported by postgraduate bursaries from the FRDŽ. SA and DAAD Ž.Germany . The second author was supported by research grants, from the University of Natal and FRD. † E-mail address: [email protected]. ‡ E-mail address: [email protected].

516

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. Ž.p,q,r-GENERATIONS OF Co3 517 groups J123, J , J , HS, and McLŽ cf.7wx᎐10. . For more information and results on the 2-generations of finite simple groups and related topics, the reader is referred towx 5 and w 15᎐17 x . This paper is devoted to the Žp, q, r.-generations and its consequences for the Conway group Co3, where p, q, r are distinct primes. We summarize our findings in the following theorem.

THEOREM 5.2. The Conway group Co3 isŽ. p, q, r -generated for all p, q, rgÄ42, 3, 5, 7, 11, 23 with p - q - r, except whenŽ.Ž. p, q, r s 2, 3, 5 . The content of this paper will be organized as follows. In Section 2 we discuss techniques that are useful in resolving generation type questions of finite groups. As a consequence of the remark in the first paragraph, we only need ot consider the cases r s 7, 11, 23 for the group Co3. We deal separately with each case in Sections 3, 4, and 5. For basic properties of the group Co3 and information on its subgroups the reader is referred to wx4, 6 . The ‘‘ATLAS of Finite Groups’’wx 3 is an important reference and we adopt its notation for subgroups, conjugacy classes, etc. Computations were carried out with the aid of GAPwx 13 running on a Sun GX2 computer. The nX-complementary generations of the group Co3 will be treated by the authors in a separate paper.

2. PRELIMINARY RESULTS

Throughout this paper we use the same notation as inwx 7᎐9 , and the reader is encouraged to consult these papers for basic computational Ž. Ž . techniques. In particular, ⌬ G s ⌬G lX, mY, nZ denotes the structure constant of G for the conjugacy classes lX, mY, nZ whose value is the cardinality of the set

⌫ s Ä4Ž.x, y g lX = mY N xy s z ,

Ž. Ž where z is a fixed element of the class nZ. Also, ⌬* G s ⌬*G lX, mY, .Ž . Ž. nZ and ⌺ H1 j ...jHr denote the number of pairs x, y g ⌫ such ²: ²: Ž . that x, y s G and x, y F Hi for some 1 F i F r , respectively. The number of Ž.x, y g ⌫ generating a subgroup H of G will be given by ⌺*Ž.Hand the centalizer of a representative of the conjugacy class lX we will denote by ClXGŽ..If⌬* Ž.G -0, then we say G is ŽlX, mY, nZ .-gen- erated and Ž.lX, mY, nZ is called a generating triple for G. We will employ results that, in certain situations, will effectively estab- lish non-generation. They include Scott’s theorem Žcf.wx 2, 14. , Ree’s theorem Žcf.wx 2, 12. , and Lemma 3.3 inwx 17 . The following result will be crucial in determining generating triples. 518 GANIEF AND MOORI

THEOREM 2.1. Let G be a finite group and H a subgroup of G containing ŽŽ. Ž. . a fixed element x such that gcd ox,wNHG :Hxs1. Then the number h of conjugates of H containing x is ␹HHŽ.x , where ␹ is the permutation character of G with action on the conjugates on H. In particular, m <

NHGŽ.. Thus the permutation character of G with this action on ⍀ is ␹Ž.1G . By definition HNsGŽH. ggx g g g ␹HGŽ.xs<<Ä4HH Ž .sH <<image of x Ž g .Žg.g ŽŽ. under the natural homomorphism NHGGªNHrH. Since ox, Ž.gg.Ž. g wNHG :Hxs1, it follows that oxs1 and hence x g H . Therefore Ž. Ä g g4 ␹H xs

G, we use the fusion map of NHGŽ.into G to determine its value on the Ž. Ž. Ž conjugacy classes. It is well known that ⌬* G s ⌬ G y ⌺ M1 j ...j Ms., where Ä4M1,...,Ms is the family of all maximal subgroups of G containing z. We also use the fusion maps in calculating ⌺Ž.Mi , for i s 1,...,s. We list in Table I the partial fusion map of the maximal subgroups M of Co3 Ž.obtained from GAP that we will use later. All the M-conjugacy classes with representatives of prime order are given. From the fusion maps we calculate the value of h Ž.see Theorem 2.1 , for x g M with oxŽ.G7.

3. Ž.p, q, 7 -GENERATION OF Co3

The group Co3 acts as a transitive rank-2 group on a set ⍀ of 276 points. The point stabilizer of this action is isomorphic to the group McL:2 Ž.p,q,r-GENERATIONS OF Co3 519

TABLE I

Partial Fusion Maps into Co3

McL:2-class 2 a 2b 3a 3b 5a 5b 7a 11a 11b ª Co3 2 A 2 B 3 A 3B 5 A 5B 7A 11 A 11B h 311 HS-class 2 a 2b 3a 5a 5b 5c 7a 11a 11b ª Co3 2 A 2 B 3B 5 A 5B 5B 7A 11 A 11B h 622

2 U4Ž.3 :2 -class 2 a 2b 2c 2 d 2 e 3a 3b 3c 5a 7a ª Co3 2 A 2 A 2 B 2 B 2 B 3 A 3B 3B 5B 7A h 3

M23-class 2 a 3a 5a 7a 7b 11a 11b 23a 23b ª Co3 2 A 3B 5B 7A 7A 11 A 11B 23 A 23B h 33 2 2 1 1

5 3:2Ž.=M11 -class 2 a 2b 2c 3a 3b 3c 3d 5a 11a 11b ª Co3 2 A 2 B 2 B 3 A 3B 3B 3C 5B 11 A 11B h 11

2иS6Ž.2 -class 2 a 2b 2c 3a 3b 3c 5a 7a ª Co3 2 A 2 A 2 B 3 A 3 A 3B 5 A 7A h 3

U33Ž.5:S-class 2 a 2b 3a 3b 3c 5a 5b 7a ªCo3 2 A 2 B 3B 3 A 3C 5 A 5B 7A h 2

4 2 и A8-class 2 a 2b 2c 3a 3b 5a 7a 7b ª Co3 2 A 2 B 2 A 3B 3B 5B 7A 7A h 33

L312Ž.4:D -class 2 a 2b 2c 2 d 3a 3b 3c 5a 7a ªCo3 2 A 2 B 2 A 2 B 3B 3B 3C 5B 7A h 1

2 = M12-class 2 a 2b 2c 2 d 2 e 3a 3b 5a 11a 11b ª Co3 2 B 2 A 2 B 2 B 2 B 3B 3C 5B 11 A 11B h 11

and the resulting permutation character is ␹McL:2 s 1a q 275a. The value of ␹McL:2 on the conjugacy class pX, p a prime, will enable us to deduce the cycle type of elements in pX as a permutation of degree 276. Ž. Ä4 LEMMA 3.1. The group Co3 is 2 X,3Y,7A-generated, for X g A, B and Y g Ä4A, B, C , if and only if the ordered pairŽ.Ž. X, Y s B, C .

Proof. If the group Co3 is Ž.pX, qY, rZ -generated, then an application of Ree’s theorem. Žcf.12towx. Co3 as a permutation group on 276 points 520 GANIEF AND MOORI

TABLE II Cycle Type of a Representative in pX

Co3-class 2 A 2 B 3 A 3B 3C 7A 11 AB ␹McL:2Ž.pX 36 12 6 15 0 3 1 Cycle type 136 2 120 1 12 2 132 1 6 3 90 1 15 3 87 3 92 1 3 7 39 1 1 11 25

implies that c123q c q c F 278, where c12, c , and c 3are the number of cycles of representatives in pX, qY, and rZ, respectively. From Table II we conclude that Ree’s theorem is violated for all triplesŽ. 2 X,3Y,7A, except when X s B and Y s C. For the tripleŽ. 2 B,3C,7A we observe from the fusion maps into Co3 that if M is a maximal subgroup with non-empty intersection with the classes in this triple, then M is isomorphic to either U33312Ž.5:S, L Ž.4:D , Ž. Ž. or S32= L 8 :3. However, we easily calculate ⌺ M s 0 for all the above Ž. Ž subgroups and hence ⌬* Co33s ⌬ Co s 504, proving the result. Ž. Ä4 LEMMA 3.2. The group Co3 is 2 X,5Y,7A-generated for X, Y g A, B , if and only if the ordered pairŽ. X, Y g ÄŽ.Ž.B, A , B, B 4. Proof. We calculate ⌬ Ž.2 A,5A,7A 21 - module V with dimŽ.VrCVVŽ.2 A s 8, dim Ž.VrC Ž.5B s 16, dim Ž.VrC VŽ.7A s 18.

But 8 q 16 q 18 - 46, and hence, by Scott’s theorem Žcf.14,2wx.ŽA,5B, 7A.is a non-generating triple of Co3. Next we consider the tripleŽ. 2 B,5A,7A. The maximal subgroups of

Co3 with order divisible by 7 and non-empty intersection with the classes 2Band 5 A are isomorphic to McL:2, HS,2иS633Ž.2 , and U Ž.5:S.We Ž. Ž . ŽŽ..Ž. calculate ⌬ Co33s 1512, ⌺ McL:2 s 0 s ⌺ U 5:S3,⌺ HS s 48, and ŽŽ.. ⌺2иS6 2s98. A fixed element of order 7 is contained in six conjugate subgroups of HS and in three conjugate copies of 2 и S6Ž.Ž2 see Table I . . Ž. Ž.Ž. Ž. Thus ⌬* Co3 G 1512 y 648 y398 )0 and therefore 2 B,5A,7A is a generating triple for Co3. Finally, we show that Co3 isŽ. 2 B,5B,7A-generated. The maximal subgroups with non-empty intersection with the classes 2 B,5B, and 7A 2 4 are, up to isomorphisms, McL:2, HS, U4338Ž.3:2, UŽ.5:S,2иA, and Ž. Ž.Ž.Ž 4 . L3124:D . We calculate ⌬ Co3s 7560, ⌺ HS s 532, ⌺ 2 и A8s 56, and ⌺Ž.M s 0 for the remaining subgroups in the above list. Also a fixed 4 element of order 7 is contained in three conjugate subgroups of 2 и A8. Ž. Ž.Ž. Thus ⌬* Co3 G 7560 y 6 532 y 356 s4200, and the result follows. Ž.p,q,r-GENERATIONS OF Co3 521

Let Ä4H1,...,Hk be the set of all non-conjugate Ž.lX, mY, nZ -generated proper subgroups of G and let z g nZ be fixed. The number of pairs Ž x, .²: yglX = mY such that xy s z and x, y + Hi, for some i, is given by k

Ýhii⌺*Ž.H ,1Ž. is1 where hiiis the number of conjugates of H containing z. Without loss of generality, assume that H1,...,Hjy1 are subgroups of Hj, where j F k. ²: g ²: g Now if x, y s Hij, for some g g G and i - j, then x, y F H . Thus from the definition of ⌺Ž.Hj it follows that kk

ÝÝhiijj⌺*Ž.HFhÝŽ.Hqh ii⌺*Ž.H. is1 isjq1 thus an understanding of the subgroup lattice of G will simplify the task of finding an upperbound for Eq.Ž. 1 .

The group Co3 acts transitively on the set ⍀ of conjugates of the subgroup M + McL. Inwx 6 the lengths of the orbits of subgroups of Co3 acting on ⍀ are determined. We shall use these orbit lengths to obtain information on the subgroup lattice of Co3. Ž. Ä LEMMA 3.3. The group Co3 is 3 X,5Y,7A-generated, for all X g A, B,C4Ä4 and Y g A, B . Proof. We will treat each triple separately.

Case Ž.3 A,5A,7A. The maximal subgroups of Co3 that have non- empty intersection with the classes 3 A,5A, and 7A are, up to isomor- Ž. Ž. Ž . phisms, McL:2, 2 и S6332 , and U 5:S. We calculate ⌬ Co3s 1680, Ž. ŽŽ.. ŽŽ.. ⌺McL:2 s 63, ⌺ 2 и S632 s 70, and ⌺ U 5:S3s0. From Table I it Ž. Ž.Ž. follows that ⌬* Co3 G 1680 y 363 y370 s1281, and hence Co3 is Ž.3A,5A,7A-generated. Ž. Ž. Case 3 A,5B,7A. We calculate the structure constant ⌬ Co3 s 175518. The maximal subgroups with non-empty intersection with the 2 classes in this triple are isomorphic to McL:2, U43Ž.3 :2 , and U Ž.5:S3.We calculate

⌺Ž.Ž.ŽMcL:2 s ⌺ McL s ⌬ McL 3a,5b,7x .s644, x g Ä4a, b , ⌺ U Ž.3:22 ⌺U Ž.3 ⌬ Ž3a,5a,7a . 112, Ž.44s Ž.s U4Ž3.s

⌺Ž.U33Ž.5:S s0.

Clearly anyŽ. 3 A,5B,7A-generated proper subgroup of Co3 is contained in either McL or U44Ž.3 . From the list of maximal subgroups of U Ž.Ž3 cf. 522 GANIEF AND MOORI

wx3.Ž. we observe that, up to isomorphisms, only L374 and A have order divisible by 3 = 5 = 7. However, ⌺ Ž.3a,5a,7a 0 ⌺ Ž.3a,5a,7a L37Ž4. s s A and hence ⌺*ŽŽ..U 3 ⌬ Ž3a,5a,7a . 112. 4 s U4Ž3. s From Lemma 4.3 inwx 9 and the above argument it is clear that, up to isomorphisms, U4Ž.3 is the only subgroup that contains Ž 3a,5b,7x .-gener- ated subgroups of McL. Furthermore, ⌺ Ž.Ž3a,5b,7x ⌬ 3a,5a, U44Ž3. s UŽ3. 7a.Žs112 the first part of the equality involves McL-classes and the Ž. . Ž . Ž . Ž . second U43 -classes . Therefore ⌺* McL s 644 y 2 112 s 420 cf.9.wx We therefore conclude that McL and U4Ž.3 are the onlyŽ. 3 A,5B,7A-gen- erated proper subgroups of Co3. It was shown by Finkelstein inwx 6 that Co3 contains a unique conjugate class of subgroups isomorphic to McL and U4Ž.3 , respectively. Therefore

⌬*Ž.Co33s ⌬ Ž.Co y 3⌺* ŽMcL .y 3⌺*Ž.U4 Ž.3 s1680 y 3Ž. 420 y 3 Ž. 112 s 84, proving generation of Co3 by this triple. Ž. Ž. Case 3B,5B,7A. We calculate the structure constant ⌬* Co3 s 175518. From the fusion maps of the maximal subgroups into Co3 we note 2 4 that McL:2, HS, U42Ž.3:2,M333831,UŽ.5:S,2 иA, and L Ž.4:D2are, up to isomorphisms, all the maximal subgroups that may containŽ 3B,5B, 7A.-generated subgroups. We calculate

⌺Ž.Ž.McL:2 s ⌺ McL s 50400, ⌺ Ž.HS s 7280, 2 ⌺Ž.U44Ž.3:2 s⌺Ž.U Ž.3 s9408, ⌺Ž.M23s 5124, 4 ⌺Ž.Ž.U33Ž.5:S s⌺U 3 Ž.5 s420, ⌺Ž.2 и A8s 420,

⌺Ž.Ž.L3123Ž.4:D s⌺LŽ.4 s882.

Thus anyŽ. 3B,5B,7A-generated proper subgroup of Co3 is contained in a 4 subgroup isomorphic to McL, HS, U423383Ž.3, M ,U Ž.5,2 иA,orLŽ.4.By investigating the maximal subgroups of these groups and their fusions into

Co3, we find that theŽ. 3B,5B,7A-generated proper subgroups of the 4 above list are, up to isomorphisms, M22,2:A 7, A 8, A 7, andŽ. if possible 4 4 subgroups of 2 и A87, other than 2 : A and A7. We list in Table III the lengths of the orbits of the above subgroups acting on ⍀. In this table mn denotes n orbits of length m.If His any subgroup of Co3 fixing at least one point MЈ g ⍀, then H F GM Ј + McL:2. Thus it follows from Table III that any McL, U4223Ž.3, M ,U Ž.Ž5 one fix 4 point on ⍀.,2:A73,LŽ.Ž.Ž.4 both classes , and A7both classes subgroup is contained in some McL:2 subgroup of Co33. It is shown inwx 6 that Co contains a unique conjugate class for each of the remaining subgroups in Table III. Ž.p,q,r-GENERATIONS OF Co3 523

TABLE III Action of H on ⍀

H Length of ⍀-orbits NHŽ. Co3

McL wx1, 275 McL:2 HS wx100, 176 HS 22 U44Ž.3wx 1 , 112, 162 U Ž.3:2 M23 wx23, 253 M23 3 U33Ž.5wx 50 , 126 U Ž.5:S3 2 U33Ž.5wx 1, 50 , 175 U Ž.5:2 M22 wx1, 22, 77, 176 M22 44 2иA88wx8, 128, 140 2 и A 4 2:A7 wx1, 7, 16, 112, 140 33 L33Ž.4wx 1 , 56 , 105 L Ž.4:D12 22 2 L3Ž.4wx 1 , 21 , 56 , 120 2 A88wx8, 15 , 70, 168 S 22 A77wx1, 7, 15 , 35 , 42, 126 S 2 A77wx1 , 7, 15, 35, 42, 70, 105 A

It therefore follows from Theorem 2.1 that the number of pairs Ž. ²: x,yg3B=5B, with xy s z a fixed element in 7A and x, y - Co3,is at most

3⌺Ž.McL:2 q 6⌺* Ž.Ž.HS q 3⌺* M23 q 2⌺*Ž.U3 Ž.5 4 q 6⌺*Ž.A88q 3⌺Ž.2 и A .2Ž.

We now proceed by finding an upperbound for the above equation. The groups A73and L Ž.4 contain no proper subgroups with order divisible by Ž. Ž. ŽŽ..ŽŽ.. 3=5=7 and hence ⌺* A77s ⌺ A s 63 and ⌺* L34 s ⌺ L 34 s 882. Up to isomorphisms, A78is the only subgroup of A that admitsŽ 3B, 5B,7A.-generation. Also a fixed element of order 7 is contained in a Ž. Ž. unique conjugate of an A788subgroup in A . Thus ⌺* A s ⌺ A8y Ž. ⌺*A7 s84 y 63 s 21. For the group U3Ž.5 we have

⌺ U Ž.5 ⌬ Ž3a,5b,7x .⌬ Ž3a,5c,7x .⌬ Ž3a,5d,7x ., Ž.3 s U333Ž5. q UŽ5. q UŽ5. where x Ä4a, b . We calculate ⌬ Ž.3a,5y,7x 140, where y Ä4b, c, d . g U3Ž5. s g Also the maximal subgroups of U3Ž.5 with order divisible by 3 = 5 = 7 are isomorphic to A7 Ž.three non-conjugate types . The fusion map of A7 into U3Ž.5 yields

3a ª 3a 3b ª 3a 5a ª 5y 7a ª 7a 7b ª 7b, 524 GANIEF AND MOORI

Ž. Ž . Ž . where y s b, c, d if A7 is of conjugate type i , ii , iii , respectively. Also a fixed element of order 7 is contained in a unique A73subgroup of U Ž.5. Ž. ŽŽ.. Thus ⌬*3U Ž5. a,5y,7x s77 and hence ⌺* U3 5 s 231. 3 Ž. Ž Next we consider the groups M22 and M23. We note ⌺ M22 s ⌬ M 3a, . Ä4 Ž. 22 5a,7x, xga,b. The 3a,5a,7x-generated subgroups of M22 are isomorphic to L37Ž.4 and A Žtwo non-conjugate copies. . Using Theo- Ž. Ž. Ž rem 2.1 we obtain ⌺* M22 s 2464 y 882 y 263 s1456. The 3B,5B, 7A.Ž-generated maximal subgroups of M23 are isomorphic to M22, L 34:2,. 2 4 2:A78, and A . From the previous arguments it follows that if H is aŽ 3B, 5B,7A.-generated proper subgroup of M23 , then H is isomorphic to 4 Ž. 4 Ž 4 either A78, A ,2:A 73,L 4, M 22,orHF2:A7. We calculate ⌺ 2: . 4 A77s336. Now 2 : A contains a subgroup isomorphic to A7, and from Theorem 2.1 we have

⌺*Ž.M23 F ⌺ Ž.M23 y 2⌺* Ž.M22 y 2⌺*Ž.L3 Ž.4 y 2⌺* Ž.A8 4 y⌺Ž.2 : A7s 70.

Ž. Ž . Ž . For the group HS we have ⌺ HS s ⌬ HS 3a,5b,7a q⌬HS 3a,5c,7a. From Lemma 2.4 inwx 9 , it follows immediately that

⌬ HSŽ.3a,5b,7a F560 y 277 Ž.Ž.y263 s260,

⌬HSŽ3a,5c,7a .F6720 y 2 Ž 1456 .y 277 Ž.y882 y 263 Ž.s2646 and hence ⌺*Ž.HS F 2906. Thus an upperbound for Eq.Ž. 2 is 170694. The Ž 3B,5B,7A .-generation Ž. of Co33follows from ⌬ Co s 175518 ) 170694. Ž. Ž. Case 3C,5A,7A. We calculate ⌬ Co3 s 85428. Up to isomor- phisms, U33Ž.5:S is the only maximal subgroup of Co3with non-empty ŽŽ. intersection with the classes of this triple. However, ⌺ U335:S s0 so that Ž.3C,5A,7A is a generating triple for Co3. Case Ž.3C,5B,7A. The maximal subgroups that contain possibleŽ 3C,

5B,7A.Ž-generated subgroups are isomorphic to U335:.S and L 3Ž4:.D 12. ŽŽ.. ŽŽ.. Ž. Ž. However, ⌺ U335:S s0s⌺ L34:D 12and hence ⌬* Co3s ⌬ Co 3 Ž. s296136, proving 3C,5B,7A-generation of Co3.

4. Ž.p, q, 11 -GENERATION OF Co3

In this section we need only to consider the maximal subgroups of Co3 with order divisible by 11. They are, up to isomorphisms, McL:2, HS, M23 , 5 32Ž.=M11 , and 2 = M12 . Ž.p,q,r-GENERATIONS OF Co3 525

Ž. Ä LEMMA 4.1. The group Co3 is 2 X,3Y,11Z-generated for X, Z g A, B4Ä4,and Y g A, B, C , if and only if the ordered pairŽ.Ž. X, Y s B, C . Proof. An application of Ree’s theorem the representatives of the classes 2 A,3B, and 11ZcfŽ.. Table II establishes that Co3 is not Ž 2 A,3B, 11Z.-generated. The action of Co3 on the 23-dimensional irreducible complex module V yields

dimŽ.VrCVVŽ.2 A s 8, dim Ž.VrC Ž.2 B s 12, dim Ž.VrC VŽ.3B s 18,

dimŽ.VrCVVŽ.3C s 12, dim ŽVrC Ž.11 AB .s 20.

Thus the triplesŽ.Ž. 2 A,3C,11Z and 2 B,3B,11Z violate Scott’s theorem, resulting in the non-generation of Co3 by these triples. Next we calculate the structure constants ⌬ Ž.Ž.2 A,3A,11Z 0 ⌬ 2B,3A,11Z and Co33s s Co non-generation by these triples is immediate. Finally, we calculate ⌬ Ž.2 B,3C,11Z 671. The maximal subgroups Co3 s of Co3 that may containŽ. 2 B,3C,11Z-generated subgroups are iso- 5 Ž. Ž5Ž.. mophic to 3 : 2 = M11 and 2 = M12. Also ⌺ 3:2=M11 s0 and Ž. Ž. Ž. Ž ⌺2=M12 s11. From Table I we conclude ⌬* Co3 s ⌬ Co3 y ⌺ 2 = . M12 s 660, proving the result.

LEMMA 4.2. The group Co3 is Ž.2 X,5Y,11Z-generated, for all X, Y, ZgÄ4A,B,except when Ž.Ž.2 X,5Y,11Z s 2A,5A,11Z. Proof. We treat the four cases separately. Ž. Ž. Case 2 A,5A,11Z. The structure constant ⌬ Co3 s 44. From the fusion maps into Co3 we note that theŽ. 2 A,5A,11Z-generated proper subgroups are contained in the maximal subgroups isomorphic to McL:2 or HS. Also ⌺Ž.Ž.McL:2 s ⌺ McL s 22 and ⌺ Ž.HS s 11. It follows from Lemmas 2.5 and 4.5 inwx 9 that no proper subgroup of McL or HS isŽ 2 A, 5 A,11Z.-generated. Thus from the fusion maps we have

⌬*Ž.Co33s ⌬ Ž.Co y ⌺* ŽMcL .y 2⌺* Ž.HS s 0,

proving non-generation of Co3 by this triple. Case Ž.2 A,5B,11Z. Every maximal subgroup with order divisible by 11 has non-empty intersection with each of the classes 2 A,5B, and 11Z. From the structure constants we calculate

⌺Ž.Ž.McL:2 s ⌺ McL s 715, ⌺ Ž.HS s 242, ⌺ Ž.M23 s 235, 5 ⌺Ž.3 :2Ž.=M11 s99, ⌺ Ž.Ž.2 = M12 s ⌺ M12 s 55. 526 GANIEF AND MOORI

Using the ATLAS and subgroup fusions into Co3, we identify all the possibleŽ. 2 A,5B,11Z-generated proper subgroups of Co3, up to iso- mophisms. They are McL, HS, M23, M 22, M 12, M 11, L 2Ž.11 , and subgroups 5 of 3 :Ž. 2 = M11 . Inwx 6 it is shown that Co3 has one conjugate class of M23, M22,M 12,M 11, and L2Ž.11 subgroups respectively. Furthermore, since 5 3:2Ž.=M11 contains M11 and L2Ž.11 subgroups, every M11 and L2Ž.11 5 subgroup of Co3 is contained in some conjugate copy of a 3 :Ž. 2 = M11 subgroup. From Theorem 2.1, it follows that the number of pairs Ž x, .²: yg2A=5B, with xy s z g 11Z a fixed element and x, y - Co3,isat most

5 ⌺Ž.McL q 2⌺* Ž.HS q 2⌺* Ž.Ž.M23 q ⌺ M12 q ⌺Ž.3 :2 Ž=M11 .Ž..3

None of the subgroups of L2Ž.11 has order divisible by 2 = 5 = 11 and Ž Ž .. Ž Ž .. Ž. hence ⌺* L2211 s ⌺ L 11 s 22. Up to isomorphisms, L211 is the only proper subgroup of M11 that isŽ. 2 A,5B,11Z-generated and a fixed element of order 11 is contained in a unique L21Ž.11 subgroup of M 1. Thus Ž. Ž. ŽŽ.. Ž. Ž. ⌺*M11 s⌺M11 y⌺*L2 11 s 11. Similarly, ⌺* M22 s ⌺ M22 y ŽŽ.. ⌺*L2 11 s 176 y 22 s 154. The onlyŽ. 2 A,5B,11Z-generated proper subgroups of each of the groups McL, HS, and M23 are isomorpic to M22, M 11, and L2Ž.11 . A fixed element of order 11Ž. in M23 is contained in a unique conjugate of an M22 , Ž. Ž. M11, and L2 11 subgroup, respectively. Thus ⌺* M23 s 253 y 154 y 11 y22 s 66. Fromwx 9 it follows immediately that ⌺*Ž.HS s 50 and ⌺*Ž.McL s 374. Thus from Eq. Ž. 3 an upperbound for the number of pairs from 2 A = 5B that produceŽ. 2 A,5B,11Z -generated proper subgroups is Ž. Ž. 762. The 2 A,5B,11Z -generation of Co33follows since ⌬ Co s 1023 ) 762. Ž. Ž. Case 2 B,5A,11Z. We calculate ⌬ Co3 s 2068. Any maximal sub- group with non-empty intersection with the classes 2 B,5A, and 11Z is isomorphic to McL:2 or HS. Furthermore, ⌺Ž.McL:2 s 0, ⌺ Ž.HS s 33, Ž. Ž. and therefore ⌬* Co3 G 2068 y 233 s2002, proving the generation of Co3 by this triple. Ž. Ž. Case 2 B,5B,11Z. The structure constant ⌬ Co3 s 7513. We ob- 5 serve from Table I that the groups isomorphic to McL:2, HS,3 :2Ž.=M11 , and 2 = M12 are the maximal subgroups of Co3 that may containŽ 2 B,5B, 11Z.Ž-generated subgroups. We calculate ⌺ McL:2.s 0, ⌺ŽHS.s 638, Ž 5 Ž..Ž. Ž. ⌺3:2=M11 s0, and ⌺ 2 = M12 s 33. Thus ⌬* Co3 G 7513 y Ž. Ž. 2 638 y 33 ) 0, proving that 2 B,5B,11Z is a generating triple of Co3. Ž.p,q,r-GENERATIONS OF Co3 527

LEMMA 4.3. The group Co3 is Ž.2 X,7A,11Y-generated, for all X, YgÄ4A,B. Ž. Ž. Proof. Case 2 A,7A,11Y. The structure constant ⌬ Co3 s 6622. TheŽ. 2 A,7A,11Y -generated proper subgroups of Co3 are contained in the maximal subgroups isomorphic to McL:2, HS, and M23. We also Ž. Ž. Ž. calculate ⌺ McL:2 s 3168, ⌺ HS s 825, and ⌺ M23 s 616. From Table I we conclude

⌬*Ž.Co3G 6622 y 3168 y 2Ž. 825 y 2 Ž. 616 ) 0,

and generation of Co3 by this triple follows. Case Ž.2 B,7A,11Y . Up to isomorphisms, McL:2 and HS are the only maximal subgroups that may admitŽ. 2 B,7A,11Y-generated subgroups. Ž. Ž . Ž. Ž. Also ⌬ Co3 s 57266, ⌺ McL:2 s 0, ⌺ HS s 2211, and hence ⌬ Co3 G 52844, proving the result. Ž. Ä LEMMA 4.4. The group Co3 is 3 X,5Y,11Z-generated, for all X g A, B,C4Ä4 and Y, Z g A, B .

Proof. Case Ž.3Z,5Y,11Z. The maximal subgroups of Co3 with order divisible by 11 and non-empty intersection with the class 3 A are isomor- 5 5 phic to McL:2 and 3 :Ž. 2 = M11 . Furthermore, a 3 :Ž. 2 = M11 subgroup does not meet the class 5 A and hence

⌬*3Ž.Ž.Ž.A,5A,11Z ⌬ 3A,5A,11Z ⌺ 3A,5A,11Z Co33G Co y McL:2 s1496 y 44 and

⌬*3Ž.Ž.Ž.A,5B,11Z ⌬ 3A,5B,11Z ⌺ 3A,5B,11Z Co33G Co y McL:2

⌺5Ž.3A,5B,11Z 1232, y3:Ž2=M11 . s

proving generation of Co3 by these triples. Case Ž.Ž.3B,5A,11Z. The 3B,5A,11Z-generated proper subgroups of Co3 are contained in the maximal subgroups isomorphic to McL:2 and Ž. Ž . Ž. HS. We calculate ⌬ Co3 s 6380, ⌺ McL:2 s 1122, ⌺ HS s 244, and Ž. hence ⌬* Co3 G 4770, proving generation. Case Ž.3B,5B,11Z. All maximal subgroups with order divisible by 11 have non-empty intersection with all the classes in the triple. Our calcula- 528 GANIEF AND MOORI tions yield

⌬*Ž.Co3G 92070 y 34485 y 2Ž.Ž. 5313 y 2 3795 y 891 y 198 s 38280, proving generation of Co3 by the tripleŽ. 3B,5B,11Z.

Case Ž.3C,5Y,11Z. The maximal subgroups of Co3 with order divisi- ble by 11 and non-empty intersection with the class 3C are up to isomor- 5 phism, 3 :Ž. 2 = M11 and 2 = M12 . However, the class 5 A does not meet either of these subgroups. Since the structure constant ⌬Co Ž3C,5A, .Ž. 3 11Z s 76472, the 3C,5A,11Z-generation of Co3 is immediate. Next, ⌬Ž.3C,5B,11Z 323081, ⌺ 5 Ž.3C,5B,11Z 1782, ⌺ Ž3C, Co31s 3:Ž2=M11. s 2=M2 5B,11Z.Ž.253. Thus ⌬*3C,5B,11Z 321046 and the generation s Co3 G of Co3 by this triple follows. Ž. Ä LEMMA 4.5. The group Co3 is 3 X,7A,11Y -generated for all X g A, B,C4Ä4 and Y g A, B .

Proof. The maximal subgroups of Co3 with order divisible by 3 = 7 = 11 are, up to isomorphisms, McL:2, HS, and M23. The subgroups HS and M23 have empty intersection with the class 3 A and therefore

⌬*3Ž.Ž.Ž.A,7A,11Y ⌬ 3A,7A,11Y ⌺ 3A,7A,11Y Co33s Co y McL:2 s22000 y 4356 ) 0. Next we calculate

⌬ Ž.3B,7A,11Y 580800, ⌺ Ž.3B,7A,11Y 132264, Co3 s McL:2 s ⌺ Ž.3B,7A,11Y 17622, ⌺ Ž.3B,7A,11Y 8272, HS s M23 s so that ⌬*3Ž.B,7A,11Y 396648. Finally, the maximal subgroups Co3 G isomorphic to McL:2, HS, and M23 do not meet the class 3C and hence ⌬*3Ž.Ž.C,7A,11Y ⌬ 3C,7A,11Y 2374614, proving the result. Co33s Co s

LEMMA 4.6. The group Co3 is Ž.5X,7A,11Y-generated, for all X, YgÄ4A,B. Proof. The maximal subgroups that may containŽ. 5, 7, 11 -generated subgroups are isomorphic to McL:2, HS, and M23. For the tripleŽ 5 A,7A, .Ž.Ž. 11Y we have 5 AF M23 s л, ⌬ Co3 s 6498712, ⌺ McL:2 s 171072, Ž. Ž. and ⌺ HS s 12672 so that ⌬* Co3 G 6302296. For the remaining case Ž. Ž.Ž. 5B,7A,11Y, we calculate ⌬ Co3 s 49618756, ⌺ McL:2 s 5132160, Ž. Ž. Ž. ⌺HS s 274593, and ⌺ M23 s 97192 and hence ⌬* Co3 ) 0, and the result follows. Ž.p,q,r-GENERATIONS OF Co3 529

TABLE IV

Structure Constants of Co3

pX 3 A 3B 3C 5 A 5B 7A 11 AB

⌬Ž.2A,pX,23Y 0 0 46 115 276 3197 7728 Co3 ⌬Ž.2B,pX,23Y 0 46 736 1955 6716 56971 120796 Co3 ⌬Ž.3A,pX,23Y ᎐ 23 529 1380 3818 33350 66700 Co3 ⌬Ž.3B,pX,23Y ᎐᎐3542 10166 44160 361284 769350 Co3 ⌬Ž.3C,pX,23Y ᎐᎐ ᎐70219 376372 2635317 4926278 Co3 ⌬Ž.5A,pX,23Y ᎐᎐ ᎐ ᎐ 817476 7893692 14954232 Co3 ⌬Ž.5B,pX,23Y ᎐᎐ ᎐ ᎐ 1106346 37913246 75202410 Co3 ⌬Ž.7A,pX,23Y ᎐᎐ ᎐ ᎐ ᎐ ᎐ 536538388 Co3

5. Ž.p, q, 23 -GENERATION OF Co3 AND MAIN RESULTS

The maximal subgroups of Co3 containing elements of order 23 are isomorphic to M23. It is evident from Table I that a fixed element of order 23 is contained in a unique conjugate of a M23 subgroup and such a subgroup has empty intersection with the classes 2 B,3A,3C, and 5 A. Thus whenever a triple Ž.pX, qY,23Z includes at least one of these Ž. Ž. classes then ⌬* Co33s ⌬ Co . Moreover, if this triple contains none of Ž. Ž. Ž. these classes, then ⌬* Co332s ⌬ Co y ⌺ M 3. Ž. LEMMA 5.1. The group Co3 is pX, qY,23Z -generated, for primes p F q and pX / qY, if and only if the ordered pairŽ. pX, qY f ÄŽ.Ž2 A,3A,2A, 3B.Ž,2B,3A .4. Proof. The result is immediate from the above remarks and Tables IV and V. 12 We summarize the main results in the following theorems.

THEOREM 5.2. The Conway group Co3 isŽ. p, q, r -generated for all p, q, rgÄ42, 3, 5, 7, 11, 23 with p - q - r, except whenŽ.Ž. p, q, r s 2, 3, 5 .

TABLE V

Struture Constants ⌺Ž.M23

pX 5B 7A 11 X

⌺Ž.2A, pX, 23Y 138 368 391 M 23 ⌺Ž.3B, pX, 23Y 2438 6624 5129 M 23 ⌺Ž.5B, pX, 23Y ᎐ 88320 61893 M 23 ⌺Ž.7A, pX, 23Y ᎐᎐135424 M 23 530 GANIEF AND MOORI

Proof. This follows from the above lemmas and the fact that the Ž. triangular group T 2, 3, 5 + A5.

COROLLARY 5.3. The Conway group Co3 isŽ. pX, pX, qY -generated, for all pX g Ä43C,5A,5B,7A,11A,11B and qY g Ä47A,11A,11B,23A,23B with p - q as well asŽ.Ž. pX, pX, qY s 3B,3B,23X . Proof. The result follows immediately from an application of Lemma 2 inwx 2 to Lemmas 3.1, 3.2, 4.1, 4.2, 4.3, and 5.1.

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