JOURNAL OF ALGEBRA 187, 579᎐619Ž. 1997 ARTICLE NO. JA966836
Verification of Dade’s Conjecture for Janko Group J3
Sonja KotlicaU
Department of Mathematics, Uni¨ersity of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
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Inwx 3 Dade made a conjecture expressing the number kBŽ.,d of characters of a given defect d in a given p-block B of a finite group G in terms of the corresponding numbers kbŽ.,d for blocks b of certain p-local subgroups of G. Several different forms of this conjecture are given inwx 5 . Dade claims that the most complicated form of this conjecture, called the ‘‘Inductive Conjecture 5.8’’ inwx 5 , will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive
conjecture for all covering groups of the third Janko group J3 Žin the notation of the Atlaswx 1. . This is one step in the inductive proof of the conjecture for all finite groups.
Certain properties of J33simplify our task. The Schur Multiplier of J is cyclic of order 3Ž seewx 1, p. 82. . Hence, there are just two covering groups of J33, namely J itself and a central extension 3 и J33of J by a cyclic group Z of order 3. We treat these two covering groups separately.
The outer automorphism group OutŽ.J33of J is cyclic of order 2 Ž seewx 1, p. 82. . In this case Dade affirms inwx 5, Section 6 that the inductive conjecture for J3 is equivalent to the much weaker ‘‘Invariant Conjecture 2.5’’ inwx 5 . Furthermore, Dade has proved inwx 6 that this Invariant Conjecture holds for all blocks with cyclic defect groups. The Sylow p-subgroups of J3 are cyclic of order p for all primes dividing < The group OutŽ 3 и JZ33< . of outer automorphisms of 3 и J centralizing Z is trivial Žseewx 1, p. 82. . In this case Dade affirms that the Inductive Conjecture is equivalent to the ‘‘Projective Conjecture 4.4’’ inwx 5 . Again, Dade has shown in wx 6 that this projective conjecture holds for all blocks with cyclic defect groups. So we only need to verify the projective conjecture for p s 2 and p s 3. We do that in Theorem 4.4.2 and Theorem 4.3.1. U This research was supported by NSF Grant DMS 93-02996. 579 0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 580 SONJA KOTLICA For the above reasons the four Theorems 2.10.1, 3.6.1, 4.4.2, and 4.3.1 below are sufficient to prove the inductive form of Dade’s conjecture for all covering groups of J3. This paper is my Ph.D. thesis. It would not have been possible without the help and infinite patience of my advisor Everett Dade. ᮊ 1997 Academic Press PART I. THE CONJECTURES In this part we explain the notation and definitions necessary to state the conjectures which we will verify in the latter part of this paper. We followwx 5 , discussing p-chains in Section 1.1 and blocks of characters in Section 1.2. The invariant conjecture is given in Section 1.3 and the projective conjecture in Section 1.4. 1.1. Radical p-Chains Let G be any finite group and let p be any prime. We will define ‘‘H F G’’ as ‘‘H is a subgroup of G.’’ We will define ‘‘H 1ᎏ G’’ as ‘‘H is a normal subgroup of G.’’ The center of G will be denoted by ZGŽ.. We will 1 1 depict conjugation exponentially so that s y and H s y H for Ž. Ž. ,gGand H F G.If H,KFG, then NHKKand CHwill be used to denote the normalizer and centralizer, respectively, of H in K. The largest normal p-subgroup of G will be denoted by OGpŽ..If Pis a p-group, then ⍀Ž.P will denote its subgroup generated by all g P p satisfying s 1. The commutator of two elements a and b of G will be 1 1 denoted by wxa, b s ababyy . The commutator of two subgroups K and 11 Lwill be denoted by wxK, L s ²klklyy P s ONpGŽ.Ž. P . Ž . Ž Ž Ž ... Since NPGGFN⍀ZP for any p-subgroup P of G, it is clear that Pis a radical p-subgroup of G if and only if it is a radical p-subgroup of NGŽ⍀ ŽZP Ž .... Hence the nontrivial radical p-subgroups of G are among the radical p-subgroups of the NHGŽ., where H runs over the nontrivial elementary abelian p-subgroups of G.If Pis a radical p-subgroup of G ŽŽ.. Ž. such that ⍀ ZP sH, then P F CHG . DEFINITION 1.1.2. A p-chain C of G is any nonempty, strictly increas- ing chain C : P012- P - P - иии - Pn DADE’S CONJECTURE 581 of p-subgroups Pi of G. The length of such a chain is < C : P01- P - иии - Pn. The stabilizer of C in any subgroup K of G is the ‘‘normalizer’’ NCKKŽ.sNP Ž0 .lNPK Ž1 .lиии l NPKn Ž .. Note that NCKKnŽ.sNC Žy1 .lNPKn Ž . if n G 1. DEFINITION 1.1.3. A radical p-chain of G is a p-chain C : P01- P - иии - Pn of G such that P0s OGpŽ. Ž. and PiGis a radical p-subgroup of NCiy1for i s 1,...,n. This last condition is equivalent to PipGis ONCŽ.Ž. for i s 1,2,..., n. We denote by R s RŽ.G the family of all radical p-chains of G. The family R is invariant under the conjugation action of G on p-chains. We denote by RrG an arbitrary set of representatives for the G-orbits in R. 1.2. Characters and Blocks Let G be a group, let H be a subgroup of G and let ރ be the field of complex numbers. We denote by IrrŽ.H the set of all irreducible ރ-char- acters of H. Since ރ is a splitting field of characteristic zero for H, the degree Ž.1 of any g IrrŽ.H divides the order < If ᑿ g BlkŽ.H induces a block ᑜ of G in the sense of Brauerw 14, p. GG 320x , we write ᑜ s ᑿ . Otherwise, ᑿ is not defined. Bywx 7, V. 5.1 the G induced block ᑿ is defined for any block ᑿ of any subgroup H F G satisfying PCGGŽ. P F H F NP Ž. Ž.1.2.1 for some p-subgroup P of G. We shall apply this when H is the normalizer NCGŽ.of some radical P-chain C : P01- P - иии - Pnof G. Ž. Ž. Ž. G Since PCnG P nFNC GFNP G n, the induced block ᑿ is defined for any block ᑿ of NCGŽ.. We are going to use several times the following well-known result. Ž. PROPOSITION 1.2.2. Assume that P is a p-subgroup of G with CG P F P 1 G. Then the principal block ᑜ is the unique p-block of G. Proof. This iswx 7, V.3.11 . PROPOSITION 1.2.3. Let H be a subgroup of G such that Ž.1.2.1 is satisfied for some p-subgroup P. Then the principal block of H induces the principal block of G. Proof. This is a consequence ofwx 7, V.6.2 . PROPOSITION 1.2.4. Two irreducible characters ijand of G belong to the same p-block if and only if G : CGiiŽ. Ž.r Ž.1'G:C Gjj Ž. Ž.r Ž.Ž1 mod ᒍ . for all elements of G, where ᒍ is a prime di¨isible by p in a suitable cyclotomic field. Proof. Seewx 7, IV.4.2 . Suppose that g IrrŽ.H for some H F G. We say that lies under a character g IrrŽ.G if is an irreducible constituent of the restriction H of to a character of H. In that case we also say that lies o¨er . We denote by IrrŽG<.Ž.the set of all g Irr G lying over . Clearly, g g g IrrŽG<.Žs Irr G< .for any g g G, where g IrrŽH .is defined by g g Žh.Ž.shfor all h g H. Now let N be any normal subgroup of G. PROPOSITION 1.2.5. The characters in IrrŽ.N lying under a fixed character gIrrŽ.G form a single G-conjugacy class. Proof. Seewx 10, V.17.3 . Ž. Ä We fix a character g Irr N , and denote by H the stabilizer h g h 4 H < s of in any subgroup H F G. Note that N 1ᎏ H whenever N 1ᎏ H. DADE’S CONJECTURE 583 PROPOSITION 1.2.6. Induction of characters is a bijection of IrrŽ H < . onto IrrŽ H < . whene¨er N 1ᎏ H F G. This bijection multiplies character degrees by the indexwx H : H of H in H. Proof. Seewx 10, V.17.5 . Clifford theory also gives us a Clifford extension for in G. This is a central extension G˜˜of some finite subgroup Z of the multiplicative group ࠻ ރ of the complex numbers by the factor group GrN. There is an exact sequence ˜˜1ᎏ 6 ⑀˜ 6 1ªZGGrNª1 in which Z˜˜is both a central subgroup of G and a subgroup of ރ࠻. Inclusion in C࠻ is a faithful linear character ˜ of Z˜˜. It follows that Z is cyclic. Let H be any subgroup of G containing N, and H˜ be the correspond- 1 ing subgroup of G˜, i.e., the inverse image ⑀˜y Ž.HrN of HrN in G˜. Clifford theory also gives us a bijection of IrrŽ H˜< ˜.Žinto Irr H < .. The exact definition of this bijection is irrelevant hereŽ seewx 4, Section 12 for details. . But that definition does give two properties of the Clifford correspondence which we can state as PROPOSITION 1.2.7. If ˜˜g IrrŽH <˜.corresponds in Clifford theory to Ž.Ž.Ž.Ž. gIrr H < , then 1 s ˜˜1 1. If, in addition, ˜g g G and g g G Ž. g˜ y1 Ž.g ha¨e the same image ˜˜⑀ g s gN in GrN, then H˜ s ˜⑀ H is the gg˜Žg˜. subgroup of G˜˜ corresponding to H F G , and the character g Irr H˜<˜ g g corresponds in Clifford theory to g IrrŽ H < .. Proof. Seewx 4, 12.14 and 12.15 . We should remark that the Clifford extension G˜ for in G is only determined to within an equivalence relation which can be defined as follows. Another central extension X ˜˜XX1ᎏ 6 ⑀˜ 6 1ªZGGrNª1 X ࠻ of a finite subgroup Z˜of ރ by GrN is equivalent to G˜if the central X X X X X products Ž.ZZ˜˜ (G˜˜˜and Ž.ZZ (G˜of G˜˜and G with the subgroup ZZ˜˜ of ࠻ X ރ are isomorphic as central extensions of ZZ˜˜ by GrN. A special situation occurs when we can extend the character g IrrŽ.N e Ž. to a character g Irr G . Ž. e Ž. PROPOSITION 1.2.8. If g Irr N extends to g Irr G , and if N F eeŽ.Ž. Ž. HFG ,then map ¬ : h ¬ hN h is a bijection of Irr HrN e onto IrrŽ H < .. This bijection multiplies character degrees by Ž.1 s Ž.1.If Ž. gN Ž.g ggG and g Irr HrN , then this bijection sends g Irr H rNto gN e e g g sŽ. g Irr ŽH < .. 584 SONJA KOTLICA Proof. Seewx 2, 11.7 . PROPOSITION 1.2.9. A character g IrrŽ.N can be extended to a charac- e Ž. e Ž. ter g Irr G if and only if it can be extended to a character q g Irr G , q for each prime q di¨iding<< GrN . Here G , q is any subgroup of G contain- ing N such that G , qrN is a Sylow q-subgroup of GrN. Proof. It is clear from Proposition 1.2.7 that can be extended to a e Ž. Ž . character g Irr G if and only if Irr G˜<˜ contains a linear character. This happens if and only if G˜˜is equivalent to a split extension of Z by GrN, i.e., if and only if we can choose the Clifford extension G˜ to be Ž. isomorphic to Z˜= GrN . Cohomology theory tells us that this happens Ž. if and only if the inverse image G˜˜q in G of a Sylow q-subgroup GrN q of GrN is equivalent to a split extension of Z˜ for each prime q dividing Ž . < PROPOSITION 1.2.10. If GrN has cyclic Sylow q-subgroups G , qrN for each prime q di¨iding<< GrN , then can be extended to a character e Ž. gIrr G . 1.3. The In¨ariant Conjecture Now we embed G as a normal subgroup in a larger finite group E.We fix an epimorphism ⑀ : E ¸ E of finite groups with kernel G. So we have an exact sequence 1ᎏ 6 ⑀ 6 1 ª GEEª1 of finite groups and their homomorphisms. For any p-chain C of G we define NCEŽ.to be ⑀ ŽNCE Ž... Since the kernel of the epimorphism Ž. Ž. Ž. Ž. ⑀:NCE ªNCE is the normal subgroup NCGEof NC, we have an exact sequence 1ᎏ 6 ⑀ 6 1 ª NCGEŽ. NC Ž. NCE Ž.ª1 of finite groups associated with C. The above group NCEGŽ.acts by conjugation on the set Irr ŽNC Ž..of all irreducible ރ-characters of its normal subgroup NCGŽ.. We denote by NCEEŽ.,the stabilizer in NCŽ.of any such , and by NCEŽ.,s⑀Ž.NCE Ž., DADE’S CONJECTURE 585 the image of that stabilizer in E. Since NCGEŽ.is contained in NC Ž, .,we have an exact sequence 1ᎏ 6 ⑀ 6 1 ª NCGEŽ.Ž. NC, NCE Ž.,ª1 ŽŽ.. associated with any p-chain C of G and any character g Irr NCG . From now on we fix a p-block ᑜ of G, an integer d G 0 and a subgroup F of E. DEFINITION 1.3.1. For any p-chain C of G we denote by kCŽ.,ᑜ,d,F ŽŽ.. the number of characters g Irr NCG satisfying G dŽ. s d, ᑿ Ž. s ᑜ , NCEŽ.,sF. The integer kCŽ.,ᑜ,d,F depends only on the G-conjugacy class of the p-chain C. So the sum in the following conjecture is well defined. Ž. Ž. The In¨ariant Conjecture 1.3.2. If OGp s1 and d ᑜ ) 0, then < The above conjecture depends on the extension E of G and the epimorphism ⑀ : E ª E as well as the p-block ᑜ of G, the integer d, and the subgroup F of E. We say that ‘‘it holds for G,’’ if it holds for the group Gand any choice of E, ⑀, ᑜ, d, and F. When ZGŽ.s1 we can identify G with the subgroup InnŽ.G of inner automorphisms in the automorphism group AutŽ.G of G. Then Conjecture 1.3.2 holds for G if and only if it holds in the special case where E s AutŽ.G and ⑀ is the natural epimor- phism of AutŽ.G onto the outer automorphism group Out Ž.G s AutŽ.G rInn Ž.G . Ž. Ž. In our case, G s J3 satisfies ZG sOGp s1. Furthermore, we may Ž. take E s Aut G which is an extension G и 2 s J3 и 2ofG. Then we may Ž. take E s Out G ( ޚ21.SoFis either ޚ or ޚ2. In order to simplify Ž.Ž. notation we write kC,ᑜ,d,i for kC,ᑜ,d,ޚi whenever i s 1, 2. So the invariant conjecture in this case is: In¨ariant Conjecture 1.3.4. If ᑜ is a p-block of G s J3 such that dŽ.ᑜ)0, if d is any nonnegative integer, and if i s 1, 2, then < 1.4. The Projecti¨e Conjecture Next we go back to an arbitrary finite group G. We pick some central extension Z и G of a cyclic group Z by G. So we fix an exact sequence 1ᎏ 6 6 1 ª ZZиGGª1 of groups such that Z is a cyclic central subgroup of Z и G. Then any p-chain C of G has a normalizer NCZиGŽ.in Z и G, the inverse image y1 ŽŽ..NCG of its normalizer in G. We fix a faithful linear character of Z in addition to the nonnegative integer d. The earlier p-block ᑜ of G is now replaced by a p-block ᑜU of ZиGlying over the block ᑿŽ. of Z containing . U DEFINITION 1.4.1. For any p-chain C of G we define kCŽ ,ᑜ ,d< . to ŽŽ..Ž. be the number of characters g Irr NCZиG <such that d s d and ᑿŽ.induces the p-block ᑜUof G. Note that kCŽ ,ᑜU,d< . depends only on the G-conjugacy class of the p-chain C. So the sum in the following conjecture is well defined. Ž. The Projecti¨e Conjecture 1.4.2. If OGp s1 and the Sylow p-subgroup U Zp of Z is not a defect group of ᑜ , then < We say that this conjecture ‘‘holds for Z и G’’ if it holds for all choices of , ᑜU, and d. PART II. THE PRIME p s 2 In this part we verify Invariant Conjecture 1.3.4 for G s J3 and the prime p s 2. In Section 2.1 we give a description of the centralizer of an involution in G, in Section 2.2 we give a description of the other maximal 2-local subgroups of G and in Section 2.3 we list the radical 2-subgroups and 2-chains of G. In the rest of the sections we analyze the normalizers of radical 2-chains leading to the verification of the Invariant Conjecture in Section 2.10. 2.1. Centralizer of In¨olution The following properties of G s J3 can be found in Janko’s paperwx 12 . The group G has only one conjugacy class of involutions. If z is an Ž. involution in G and H s CzG , then bywx 12, Lemma 2.13 , the group H is DADE’S CONJECTURE 587 a semi-direct product 0 H s E i A5. Ž.2.1.1 Here E is an extra special group of order 2 5 which is the central product of a dihedral group of order 8 and a quaternion groupwx 12, Lemma 2.2 . 0 Also A5 is an alternating group of degree 5 acting faithfully on E. The Ž. Ž. ²: centralizer CEG is ZEs z bywx 12, Lemma 2.1 . 0 We can describe the structure of this semi-direct product E i A5 quite explicitly. Let ⍀ be the 5-element setÄ4 1, 2, 3, 4, 5 and S5 be the symmetric group of degree 5 operating on ⍀. We begin by forming the Clifford algebra C5 Žseewx 15. , which is the associative algebra over ޑ generated by a unity 1 and elements ei, for each i g ⍀, subject only to the relations 2 ei sy1, eeijsyee ji if i / j 5 for any i, j g ⍀. Then the algebra C5 has dimension 2 and a basis Ä41, e123451213, e , e , e , e , ee,ee,...,ee 45123,eee,...,eee 345,... , consisting of all products ee иии e , where 1 i - i - иии - i 5 ii12 ik F 12 kF and k G 0. Look at the subgroup ⌫ of the unit group of this algebra generated by the ei for i g ⍀. The elements of ⌫ are the products "ee иии e , where k 0 and 1 i - i - иии - i 5. The order of ii12 ik G F 12 kF ⌫is 26 . The center ZŽ.⌫ is the cyclic group of order 4 generated by 2 Ä4 sseeeee12345. Note that s sy1. The quotient group ⌫ s ⌫r "1is 5 elementary abelian of order 2 . It is generated by the images e i of the elements eiifor 1 F i F 5. These generators e satisfy the conditions 2 eiis1,eejjseeifor any i, j g ⍀. The subgroup of ⌫ generated by s s eeeee12345 has a unique S5- ² :²: invariant complement eeij<1Fi-jF5in⌫. The inverse image of s Ž. ² : in ⌫ is Z ⌫ . The inverse image of eeij<1Fi-jF5 is the subgroup E˜ of ⌫ given by E˜sÄ4"1jÄ4"eeij<<1Fi-jF5 jÄ"eeeijkl e1Fi-j-k-lF5.4 Ž.2.1.2 5 The group E˜is an A5-invariant extra special group of order 2 such that Ž. Ž. Ä4 E˜˜rZE is elementary abelian and ZE˜s"1 . In fact, E˜is S5- invariant. 588 SONJA KOTLICA 0 We can identify E with E˜ and H s E i A55with E˜i A . Then z is identified with y1. Since z is the only involution in ZEŽ., the normalizer of E in G is 0 NEGGŽ.sCz Ž.sHsEiA5. Ž.2.1.3 Ž. PROPOSITION 2.1.4. The extension group NGи2 E s H и 2 is isoclinic to E˜˜iA555и2sEiS,where S acts on elements in E˜ by permuting their indices as abo¨e. Proof. In order to prove that H и 2 and E˜i S5 are isoclinic we have to Ž. Ž .Ž provide two isomorphisms, one of H и 2rZHи2 onto E˜˜i S5rZEi S55.and one of wxwxH и 2, H и 2 onto E˜˜i S , E i S5, such that the action of Hи2rZHŽ.и2onwxHи2, H и 2 induced by conjugation in H и 2 is carried Ž.Ž. into the action of E˜˜i S55rZEiS on wxE ˜˜i S 55, E i S induced by conjugation in E˜i S5 Žseewx 1, xxiii. . Ž.Ž.²: Ž .Ž. We know that ZHи2sZEs z and ZE˜˜iS5 sZEs ²: 0 "1 . Furthermore wxwxH и 2, H и 2 s H, H s H s E i A5 and wE˜i 0 S55,E˜˜iSxsEiA 5. We have already identified H s E i A5with E˜i 0 0 A55so that E s E˜and A s A5. This provides the isomorphism of wH и 2, H и 2xw onto E˜i S55, E i S x. So the only problem is to construct a Ž.²: ²: suitable isomorphism of E˜i S5 r " 1 onto H и 2r z . Of course, Ž.²: E˜iS5 r"1 is isomorphic to the semi-direct product E i S5, where EsE˜r²:"1sEr ²:z. Evidently, must agree with the identification Ž.²: ²: 0 of E˜i A55r " 1 s E i A with Hr z s E i A 5. We want to define Ž. on the generator s 12 of S55modulo A . We know fromwx 1, p. 82 that H и 2 is an extension of E by the group S5. Hence there is an epimorphism of H и 2 onto S5 with kernel E such Ž. 0 that a55s a for any a 555g A s A . We may pick an element g H и 2 yHso that Ž.s. The four group V s Ä1,Ž.Ž.Ž.Ž.Ž.Ž. 12 34 , 13 24 , 14 234 is a Sylow 2-subgroup 0 q Ä4 Ä of A5. Let E be E s Er "1 written additively. Then ee15, q ee25,ee 35,ee 454form a ޚ2-basis for E . The group V acts regularly and transitively on this basis. So q E(ޚ12wxV as a ޚ wxV -module.Ž. 2.15 n It follows that the cohomology group HVŽ,Eq.is zero for all n G 1. q nŽ 0 q. Since 2E s 0, the cohomology group HA5,E is equal to its Sylow n 0 q n q 2-subgroup HAŽ 52,E. , which is isomorphic to a subgroup of HVŽ ,E. nŽ0q. s0 bywx 10, I.16.19 and 16.21 . Hence HA5,Es0 for all n ) 0. Ž. Ž. 0 The factor group NEG rZE is a semi-direct product E i A 5, where 00Ž. Ž. 1Ž0 q. A55is the isomorphic image of A in NEGrZE. Since HA5,E s0, 0 there is only one E-conjugacy class of complements to E in EA 5. Conjuga- DADE’S CONJECTURE 589 Ž. Ž. tion by the element in NEGи2 yNEG permutes the complements of 0 E in E i A 5. Hence we can multiply by an element e of E so that e 0 Ž. Ž. normalizes the fixed complement A 5. Since e s s, we may 0 replace by e and assume that normalizes A55s A . This normal- 00 0 izes the inverse image ZEŽ.=A55of A in NEGŽ..So ŽA5.is another X 00 complement A55to ZEŽ.in ZE Ž.=A. Since A5is perfect group, Ž. 0 Ž. XX00 ZE=A555sZE=A implies A s A5. Hence leaves A 5invariant. Because is the identity on A5, this implies that conjugation by in H и 2 0 agrees with conjugation by in E˜i S55on the common subgroup A s A5 of these two groups. Now the map 1 y : x ¬ Ž.x 0 0 is an automorphism of E i A55s E˜i A which is the identity on A55s A . 0 We claim that is the identity map of E i A5 onto itself. 0 For any a55g A and e g E, we have a y1 5 aa Ž.esŽ.e55sŽ.e y1 Ž. Ž 0. since a55s a . Evidently, sends ZEsZEiA5onto itself. So it Ž0.Ž. 0 induces an automorphism of E i A55rZEsEiA acting as the 0 q identity on A 5. It follows that restricts to an automorphism ␣ of E as 0 q 0 a ޚ25A -module. But E is an absolutely irreducible ޚ25A -module. So the q unit ␣ of End 0ŽE .ޚ must be the identity map of E onto itself. A 5 ( 2 0 Therefore is the identity map of E i A 5 onto itself. Now restricts to an automorphism ␣ of E such that ␣ acts as the identity on both ZEŽ.and E s ErZE Ž.. Bywx 10, I.17.1 there is a homo- morphism : E ª ZEŽ. such that Ž.e s e Ž.eZ Ž E . 0 for any e g E. Because is the identity on A5, the map must be a 0 q qq q ޚ25A-homomorphism of E into ZEŽ.. Since E and ZEŽ.are noniso- 0 morphic simple ޚ25A -modules, must be trivial. Hence is identity on E 0 0 and on A55. So it is identity on E i A . 22 2 Since s 1, it follows that g H centralizes H.So gZHŽ.s ²:z is either z or 1. In both cases the image of in Ž.²:H и 2 r z satisfies 2 ²: s1 and has the same conjugation action as on Hr z s E i A5.It Ž.²: follows that we can extend from E i A55to E i S s E i A 5i z by Ž. setting s. This extended preserves the actions of E i S5 and Ž.²: 0 Hи2rzon H s E i A55s E˜˜i A . Hence the groups H и 2 and E i S5 are isoclinic. 590 SONJA KOTLICA ²: COROLLARY 2.1.6. The factor group H и 2r z is isomorphic to E i S5. Proof. This is an immediate consequence of the proposition. 2.2. Other Maximal 2-Local Subgroups There are three conjugacy classes of involutions in H. The involutions in HyElie in a single conjugacy class in H bywx 12, Lemma 2.4 . The noncentral involutions in E form a single conjugacy class in H by the Ž.Ž. 0 second sentence in the proof ofwx 12, Lemma 2.3 . We fix 12 34 g A5,an involution from H y E. Also we fix the four subgroup S sy²1, Ž 12 .Ž 34 .: of H. We fix eeee1234, a noncentral involution in E and also the four-sub- ²: 0 group R sy1, eeee1234 of E. Clearly, the alternating subgroup A4on 0 0 Ä41, 2, 3, 4 of A5 centralizes R.SoCRHEŽ.contains CR Ž.iA4. By lines 5᎐7 of the proof ofwx 12, Lemma 2.3 , the index A s CHŽ.Ž.Ž.12 34 sÄ"1," eeee1234,"eeee 1234,"eeee 1234␥,"eeee 1234␦, ","␥,"␦4,Ž. 2.2.1 where  s Ž.Ž.12 34 , ␥ s Ž.Ž.13 24 , and ␦ s Ž.Ž.14 23 . Then B becomes the centralizer B s CeeHŽ.12Ž.Ž.12 34 sÄ"1," eeee1234,"ee 12,"ee 34,"ee 13␥,"ee 24␥, "ee14␦,"ee 23␦4.Ž. 2.2.2 We can conclude that ²: AlBsy1, eeee1234 sR, ABrŽ.Ž.Ž.A l B s Ar A l B = Br A l B , Ž. 2.2.3 6 and AB is a subgroup of order 2 in CRGŽ.. But the unique Sylow Ž. 6 2-subgroup T1 of CRG sT11iQalso has order 2 . So T s AB is a Ž. Ž. special 2-group. Furthermore, CRGHsCRsAB i Q is isomorphic DADE’S CONJECTURE 591 to the group of all matrices 100 ai 0, 0bc1 2 where a, b, c g GFŽ. 4 s Ä0, 1, , 4and i s 1, 2, 3. Under this isomor- phism T1 is mapped onto the group of all matrices 100 a10, ž/bc1 where a, b, c g GFŽ. 4 , and Q onto the group of all matrices 100 0i 0, 0001 where i s 1, 2, 3. Bywx 12, Lemma 3.1 the normalizer NTGŽ.1 satisfies ²: NTGŽ.1sNRG Ž.sT1i ŽPi ., where PŽ.> Q is an elementary group of order 9 normalized by an involution . We can fix to be s eeee1235.Ž. 2.2.4 ²Ž .: Ž. Then Q s 123 is centralized by . In fact Q s CP . There are four ŽŽ distinct subgroups of order 3 in P. Three of them are P1s CAP rAl .. Ž Ž .. Ž. B , P2 s CBPPrAlB, and Q s CAlB. Let P4be the fourth subgroup of order 3 in P. Then by line 16 of the proof ofwx 12, Lemma 3.1 , 2 ²: the group S33s P is isomorphic to S3. It follows that conjugation by interchanges P12and P and, hence, interchanges A and B. We conclude that the normalizer of T1 s AB is ²: 2 NABGŽ.sAB i Ž.Q = P4333 s AB i Ž.Q = S ( AB i Ž.ޚ = S . Ž.2.2.5 PROPOSITION 2.2.6. We ha¨e 1 NAGŽ.sAiŽ.P25=A(AiŽޚ 35=A .(AiGL 2 Ž. 4 , Ž. Ž.4 where GL22 4 acts on A ( ޚ as it acts on the two-dimensional ކ4-¨ector space. 592 SONJA KOTLICA Ž. Ž. Proof. We have CAGHsCAsA. Since G has only one conjugacy class of involutions, we know fromwx 12, Lemma 3.3 that the group Ž. Ž. NAG rAis isomorphic to GL235 4 ( ޚ = A and NAGŽ.(Ai Žޚ35=A .(AiGL 2 Ž. 4 , Ž 2.2.7 . 2 where GL24Ž. 4 acts on A as it acts on Ž.ކ . Since P normalizes A and has 2 order 3 , it follows fromŽ. 2.2.7 that P is a Sylow 3-subgroup of NAGŽ.. Ž. Ž . Ž . Hence the first isomorphism i : NAG ªAiޚ35=Ain 2.2.7 can be Ž. chosen so that ޚ3 = 1 F iP. This isomorphism must send the 2-subgroup Ž. Ž.Ž. AB of NAG into a 2-subgroup iAB˙ sAi1=V, where V is a four-subgroup of A53. Evidently ޚ = 1 centralizes V and, hence, it central- Ž. y1Ž. izes iABrA. It follows that i ޚ3 = 1 is a subgroup of order 3 in P centralizing ArŽ.A l B ( ABrA. From the above discussion of the sub- y1 y1 groups of P, this forces i Ž.ޚ32= 1tobeP. Since i Ž.1 = A 5is a copy 1 Ž. Ž 1. A55of A , we conclude that NAGsAiP25=A. 2.3. Radical 2-Subgroups and 2-Chains We first list the radical 2-subgroups of G. PROPOSITION 2.3.1. The group G has exactly fi¨e conjugacy classes of radical 2-subgroups, represented by 1, A, AB, E, and T. Proof. The above listed groups satisfy: NGŽ.1 s GOG2 Ž .s1 1 NAGŽ.sAiŽ.P25=A(AiGL 2Ž. 4 OA 2Ž.iGL 2Ž. 4 s A 2 2 NABGŽ.sAB i Ž.Q = S32OABŽ.i Ž.Q=S 3sAB 0 0 NEGŽ.sEiA52OEŽ.iA 5sE NTGHŽ.sNT Ž.sTiQOT2Ž.iQsT So they are all radical 2-subgroups of G. No two of them are conjugate because no two of them are isomorphic. Let W ) 1 be a radical 2-subgroup of G. Then Y s ⍀ŽŽZW ..is a Ž. Ž. nontrivial elementary abelian 2-subgroup of G with NWGGFNY.By wx8, Lemmas 2.1᎐2.3 , the normalizer NYGŽ.is G-conjugate to a subgroup of K, where K is one of NAGGŽ.,NAB Ž .,orNE G Ž.. So we may replace W Ž. by a G-conjugate and assume that NWG FK. Then W is a radical 2-subgroup of K. DADE’S CONJECTURE 593 Ž. Ž. Suppose that K s NAG (AiGL2 4 . Then W ( A i U, where U is a radical 2-subgroup of GL22Ž. 4 . But the Sylow 2-subgroups of GLŽ. 4 are trivial intersection subgroups. So the radical 2-subgroups of GL2Ž. 4 are 1 and the Sylow 2-subgroups. Hence W is conjugate to either A or AB in this case. Ž. Ž. Suppose that K s NABGG. The only radical 2-subgroups of NAB are AB and the Sylow 2-subgroups of NABGŽ..SoW is conjugate to either AB or T in this case. Ž. Ž. Suppose that K s NEGG. Since the Sylow 2-subgroups of NErE( A5 are trivial intersection subgroups, the only radical 2-subgroups of NEGGŽ.are E and the Sylow 2-subgroups of NEŽ..SoW is conjugate to either E or T in this case. Since W is conjugate to one of A, AB, E,orT in any case, the proposition is proved. Next we describe radical 2-chains of G and their normalizers in G. PROPOSITION 2.3.2. There are exactly eight conjugacy classes of radical 2-chains of G. These radical chains are represented by the following chains: C0,1 :1 NCGŽ.0,1 sG ²: C1,1 :1-AB NGŽ. C1,1s AB i Ž.Q = P 4 ( AB i Ž.ޚ 3= S 3 1 C1,2 :1-ANCGŽ.1,2sAiŽ.P 2=A 5(AiGL 2Ž. 4 C2,1 :1-A-AB NGŽ. C2,1s AB i ŽQ = P 4 .( AB i Žޚ 3= ޚ 3 . 0 C1,3 :1-ENCGŽ.1,3sEiA 5sE˜iA 5 ²: C2,2 :1-E-TNCGŽ.2,2 sAB i Ž.Q = s T i Q ²: C2,3 :1-AB - TNCGŽ.2,3 sAB i Ž.Q = s T i Q ²: C1,4 :1-TNCGŽ.1,4 sAB i Ž.Q = s T i Q Proof. If a radical chain has length 0, it is 1 and its normalizer is G.Ifa radical 2-chain in G has length 1, it is 1 - A,1-AB,1-E,or1-T by Proposition 2.3.1. If it has length 2, it must start with a radical 2-chain 1 - X of G followed by X i Y, where Y is a radical 2-subgroup of a complement to X in NXGŽ.. 1 The only radical 2-subgroups of the complement P25= A to A in 11 NAGŽ.are 1 and the Sylow 2-subgroups V of A5. Hence, if a chain starts with 1 - A, it must be followed by A i V 1 which is conjugate to AB. The only radical 2-subgroups of the complement Q = P4²: to AB in NABGŽ.are 1 and the conjugates of ²: . If a chain starts with 1 - AB,it Ž. ²: must be followed by an NABG -conjugate of AB i ( T. 594 SONJA KOTLICA 0 The only radical 2-subgroups of the complement A5 to E in NEGŽ.are 0 0 1 and the Sylow 2-subgroups V of A5. So if a chain starts with 1 - E,it must be followed by E i V 0 which is conjugate to T. Ž. The only radical 2-subgroup of the complement Q ( ޚ3to T in NTG is 1. So if a chain starts with 1 - T, it stops there. 2.4. The Normalizer of the Radical 2-Chain C0, 1 For each radical 2-chain Ci, j in Proposition 2.3.2 we shall compute the numbers kNŽŽGi C,j .,ᑜ,d,i .Žin 1.3.5 . . We start with the chain C0, 1. LEMMA 2.4.1. There are fi¨e 2-blocks of G. The set of irreducible charac- ters of G consists of 17 characters with degrees 1, 85, 85, 323, 323, 324, 646, 646, 816, 1140, 1215, 1215, 1615, 1938, 1938, 2754, and 3078 which belong to the principal 2-block ᑜ and four characters with degrees 1920, 1920, 1920, and 2432 which lie in separate 2-blocks of defect zero. The action of G и 2 interchanges the pairs of characters with degrees 85, 323, 646, and 1938 and fixes the remaining characters. Hence the following holds: kNŽ.Ž.GŽ. C0,1 ,ᑜ,7,2 s4, kNGŽ. C0,1 ,ᑜ,7,1 s4 kNŽ.Ž.GŽ. C0,1 ,ᑜ,6,2 s2, kNGŽ. C0,1 ,ᑜ,6,1 s4 kNŽ.Ž.GŽ. C0,1 ,ᑜ,5,2 s2, kNGŽ. C0,1 ,ᑜ,5,1 s0 kNŽ.Ž.GŽ. C0,1 ,ᑜ,4,2 s0, kNGŽ. C0,1 ,ᑜ,4,1 s0 kNŽ.Ž.GŽ. C0,1 ,ᑜ,3,2 s1, kNGŽ. C0,1 ,ᑜ,3,1 s0. ŽŽ . . The integer k NG C0, 1 , ᑜ, d, i is zero for all other ¨alues of d and i. Proof. Use Proposition 1.2.4. and the character table of J3 in the Atlas. 2.5. The Normalizer of the Radical 2-Chain C1, 1 In this section we consider the radical 2-chain C1, 1 :1-AB whose normalizer in G is the group ²: NABGŽ.sAB i Ž.Q = P433 ( AB i Ž.ޚ = S . Bywx 1 the normalizer of AB in G и 2 is a group of the form AB i Ž.S33= S . So it must satisfy ²: ²: NABGи2433Ž.sAB i Ž.Q = P ( AB i Ž.S = S , where is an element of order 2 lying in G и 2 y G. DADE’S CONJECTURE 595 LEMMA 2.5.1. The principal 2-block ᑜ1, 1 is the only 2-block of NGŽ. C1, 1 . It induces the principal block ᑜ of G. It contains 20 irreducible characters 6323 6 whose degrees form the multiset Ä1 , 2 , 9 , 6 , 124 . The normalizer NGи21,1Ž. C fixes two characters of degree 1, one character of degree 2, one character of degree 6, two characters of degree 9, and two characters of degree 12. It mo¨es the remaining characters. The following holds: kNŽ.Ž.GŽ. C1,1 ,ᑜ,7,2 s4, kNGŽ. C1,1 ,ᑜ,7,1 s4 kNŽ.Ž.GŽ. C1,1 ,ᑜ,6,2 s2, kNGŽ. C1,1 ,ᑜ,6,1 s4 kNŽ.Ž.GŽ. C1,1 ,ᑜ,5,2 s2, kNGŽ. C1,1 ,ᑜ,5,1 s4. ŽŽ . . The integer k NG C1, 1 , ᑜ, d, i is zero for all other ¨alues d and i. Ž. Ž. Proof. Since CABG sZAB, Proposition 1.2.2 says that the princi- pal 2-block ᑜ1, 1 is the only 2-block of NCGŽ.1, 1 . By Proposition 1.2.3 it induces the principal 2-block of G. The degrees of the irreducible charac- 2 ²: ters of S34s P are 1, 1, and 2, and those of Q are 1, 1, and 1. Thus there are nine irreducible characters of NABGŽ.which are trivial on AB. Of these nine characters six have degree 1 and three have degree 2. 2 Conjugation by fixes all irreducible characters of S3 , interchanges 2 irreducible characters of Q, and fixes the remaining one. So NCGи21,1Ž. fixes exactly two irreducible characters of NABGŽ.of degree 1 and one of degree 2. The nine linear characters of ABrŽ.A l B nontrivial on both Ar ŽA l .Ž. 2 Band Br A l B form a single Q = S3 -orbit. Some character in this orbit has ²: as its stabilizer in Q = P4²: and ²: = ²: as its stabilizer in Q²: = P4 ²: . Hence lying over these nine characters there are two irreducible characters of NABGŽ.with degree 9, both of which are invariant under NCGи21,1Ž.. The six linear characters of ABrŽ.A l B trivial on one of ArŽ.A l B or BAŽ.lB and nontrivial on the other form a single orbit under 22 Q=S33. The subgroup S acts regularly on them. The stabilizer in Q²: ²: 32²: =P43 of one of them, say , is a complement S of S3s P4 and, ²: 12 hence, is isomorphic to Q s S33. The stabilizer of in Q = S is a 3 cyclic group A3 of order 3 whose generator is inverted by an automor- 3 3 2 phism 33g S y A 3and whose index in Q = S3is 6. So lying over these six characters there are two irreducible characters of NABGŽ.of degree 6 interchanged by NABGи2Ž., and one character of degree 6 fixed by NABGи2Ž.. 596 SONJA KOTLICA The irreducible characters of AB nontrivial on ZABŽ.sAlBare 0onAB y A l B ⌳ i s ½ 4i on A l B, where i, for i s 1, 2, 3, are the distinct nontrivial linear character of 2 A l B. The characters 123, , are permuted by A3s P 4regularly and transitively. One of them, say 11, is fixed by . The stabilizer of in 12 2 1 S33=S is a complement to A3containing . Hence it is S3= ²: and its intersection with G is Q = ²: . e LEMMA 2.5.2. We claim that we can extend ⌳11to a character ⌳ of the Ž1²:. Ž ²: ²:. Ž. stabilizer AB S31= s AB Q = of ⌳ in NGи2 AB . Proof. To prove this it suffices to show that ⌳1 can be extended to a Ž 1 ²:. character of the inverse image of a Sylow p-subgroup of AB S3 = rAB 1 ²:Ž. for each prime p s 2, 3 dividing there are six irreducible characters of NABGŽ.of degree 12, of which two are fixed by NABGи2Ž.and four are not. This completes the proof of Lemma 2.5.1. 2.6. The Normalizer of the Radical 2-Chain C1, 2 The normalizer in G of the radical 2-chain C1, 2 ,1-Ais 1 NAGŽ.sAiŽ.P25=A(AiGL 2Ž. 4 . Bywx 9, p. 90 the normalizer of C1, 2 in G и 2is 1 ²: ²: NAGи2Ž.sAiŽ.P 2=A 5i 1,2(AiGL 2Ž. 4 i 1,2 (Ai⌫L42Ž. for some element 1, 2 of order 2 such that conjugation by 1, 2 acts on Ž. 1 GL225 4 ( P = A as the Galois automorphism. LEMMA 2.6.1. There is only one 2-block ᑜ1, 2 of NGŽ. C1, 2 . It induces the principal block ᑜ of G. The set of irreducible characters of NGŽ. C1, 2 consists of 19 characters. The multiset of their degrees is Ä1,3,4,5,15,45.3633 3 4The normalizer NGи21,2Ž. C fixes one character of each degree 1, 4, 5, 15, and 45, and mo¨es all other characters. Hence the following holds: kNŽ.Ž.GŽ. C1,2 ,ᑜ,6,2 s4, kNGŽ. C1,2 ,ᑜ,6,1 s12 kNŽ.Ž.GŽ. C1,2 ,ᑜ,5,2 s0, kNGŽ. C1,2 ,ᑜ,5,1 s0 kNŽ.Ž.GŽ. C1,2 ,ᑜ,4,2 s1, kNGŽ. C1,2 ,ᑜ,4,1 s2. ŽŽ . . The integer k NG C1, 2 , ᑜ, d, i is zero for all other ¨alues d and i. Ž. Proof. Since A s CAG , Proposition 1.2.2 says that the only 2-block ᑜ1, 2 of NCGŽ.1, 2 is the principal block. By Proposition 1.2.3 it induces the 1 principal block ᑜ of G. The irreducible characters of A5 have degrees 1, 5, 3, 3, and 4, and those of P2 have degrees 1, 1, and 1. So there are irreducible characters of NAGŽ.of degrees 1, 1, 1, 5, 5, 5, 3, 3, 3, 3, 3, 3, 4, 4, and 4 lying over the trivial character of A. Conjugation by 1, 2 acts on P23( ޚ by inversion. So it fixes one character of P2and moves the other 1 two. It also acts as a noninner automorphism of A5. So it fixes the 1 characters of A5 with degrees 1, 4, and 5 and moves the other two with degree 3. Hence the normalizer NCGи21,2Ž.fixes one character of each Ž. degree 1, 4, and 5, and does not fix any other character of NAG rA. The 15 nontrivial linear characters of A form a single GL2Ž. 4 -orbit. The stabilizer in GL24Ž. 4 of any one of them is isomorphic to A , and in ⌫L42Ž. is isomorphic to S4. Hence lying over these 15 characters there are 598 SONJA KOTLICA irreducible characters of degrees 15, 15, 15, and 45. Two of the characters of degree 15 are interchanged by NAGи2Ž., while the other character of degree 15 and that of degree 45 are fixed. 2.7. The Normalizer of the Radical 2-Chain C2, 1 The normalizer of the chain C1, 2 :1-A-AB in G is i 1 0 AB i Ž.Q = P424s A i Ž.P = A ( A i , ½5ž/aj 2 where a g GFŽ. 4 s Ä0, 1, , 4and i, j s 1, 2, 3. In G и 2 the normalizer of C2, 1 is ²: ²: AB i Ž.Q = P42,1242,1i s A i ŽP = A .i i 0²: (Aii2,1 , ½5ž/aj where conjugation by 2, 1 acts as the Galois automorphism 2,1 i 0 2i 0 s . ž/ž/aj a22j for all a g GFŽ. 4 and i, j s 1, 2, 3. LEMMA 2.7.1. There is only one 2-block ᑜ 2, 1 of NGŽ. C2, 1 . It induces the principal 2-block ᑜ of G. The set of irreducible characters of NGŽ. C2, 1 consists of 19 characters. The multiset of their degrees is Ä196 , 3 , 9, 12 34 . The normalizer NGи22,1Ž. C fixes one character of degree 1, two characters of degree 3, one character of degree 9, and one character of degree 12, and mo¨es all the other characters. The following holds: kNŽ.Ž.GŽ. C2,1 ,ᑜ,6,2 s4, kNGŽ. C2,1 ,ᑜ,6,1 s12 kNŽ.Ž.GŽ. C2,1 ,ᑜ,5,2 s0, kNGŽ. C2,1 ,ᑜ,5,1 s0 kNŽ.Ž.GŽ. C2,1 ,ᑜ,4,2 s1, kNGŽ. C2,1 ,ᑜ,4,1 s2. ŽŽ . . The integer k NG C2, 1 , ᑜ, d, i is zero for all other ¨alues of d and i. Ž. Ž. Proof. Since CABG sZAB, Proposition 1.2.2 says that the only 2-block of NCGŽ.2, 1 is the principal block. By Proposition 1.2.3 it induces the principal 2-block ᑜ of G. The degrees of the irreducible characters of P24are 1, 1, and 1, and of those of A are 1, 1, and 3. Hence there are nine irreducible characters of NCGŽ.1, 2 of degree 1, and three irreducible DADE’S CONJECTURE 599 characters of degree 3, lying over the trivial character of A. Since conjuga- tion by 2, 1 is a noninner automorphism of P2 and of A4 , it moves eight of these characters of degree 1 and two characters of degree 3, and leaves fixed one of these characters of each degree 1 and 3. The group P2 permutes transitively and regularly the three nontrivial linear characters of A which are trivial on A l B. The stabilizer in 1 1 P24= A of any such character is isomorphic to A4. Its stabilizer in Ž1.² : 1 ²: P2=A 4i 2, 1 is isomorphic to A4i 2, 1( S 4. So lying over these characters of A there are irreducible characters of NCŽ.2, 1 of degrees 3, 3, 3, and 9. Two of them of degree 3 are interchanged by NCGŽ.2, 1 and the remaining two are fixed by NCGŽ.2, 1 . Ž. Ž. The irreducible characters of NC2, 1 nontrivial on ZABsAlBlie over the nonlinear characters of AB, 0onAB y A l B ⌳ i s ½ 4i on A l B, where i, for i s 1, 2, 3, are the nontrivial irreducible characters of A l B. The group P41permutes , 2, 3regularly and transitively. The stabilizer in Q = P41of is a complement to P4. e Let 11be an extension of to A. No nontrivial element in the group e Ž. ABP41rA stabilizes . Let X be the normalizer NGи21 - A - AB s AB e iŽ.²:Q=P42,1i. The stabilizer Xeof 1contains A. The factor Ž.1 Ž Ž . group XerA is a complement to ABP44rA in AB i Q = P i ²:.1 2, 1 rAand, hence, is isomorphic to S3. Since all the Sylow subgroups of e Sare cyclic, can be extended to X eŽ.see Proposition 1.2.10 . It follows 31 1 e that there are three extensions of to A i Ž.Q = P e. One of these 141 characters is fixed by X e ; the other two are interchanged. So lying over 1 the orbit of ⌳12, ⌳ , and ⌳ 3there are three irreducible characters of NCŽ.2,1 of degree 12, two of which are interchanged by NCGи22,1Ž.. 2.8. The Normalizer of the Radical 2-Chain C1, 3 The normalizer of the chain C1, 3 in G is 0 NEGŽ.sEiA55sE˜iA Ž. as defined in Section 2.1. Its normalizer in G и 2is NEGи25sEиSwhich is isoclinic to E˜i S5 by Proposition 2.1.4. LEMMA 2.8.1. There is only one 2-block ᑜ1, 3 of NGŽ. E . It induces the principal 2-block ᑜ of G. The set IrrŽŽ..NG E consists of sixteen characters. Their degrees form the multiset Ä1, 32422 , 4, 5 , 8 , 10 , 15, 16, 20, 244 . The nor- malizer NGи2Ž. E fixes one character of each degree 1, 4, 15, 16, 20, or 24, and 600 SONJA KOTLICA two characters of each degree 5 or 10. It mo¨es all the other six characters. The following holds: kNŽ.Ž.GŽ. C1,3 ,ᑜ,7,2 s4, kNGŽ. C1,3 ,ᑜ,7,1 s4 kNŽ.Ž.GŽ. C1,3 ,ᑜ,6,2 s2, kNGŽ. C1,3 ,ᑜ,6,1 s0 kNŽ.Ž.GŽ. C1,3 ,ᑜ,5,2 s2, kNGŽ. C1,3 ,ᑜ,5,1 s0 kNŽ.Ž.GŽ. C1,3 ,ᑜ,4,2 s1, kNGŽ. C1,3 ,ᑜ,4,1 s2 kNŽ.Ž.GŽ. C1,3 ,ᑜ,3,2 s1, kNGŽ. C1,3 ,ᑜ,3,1 s0. ŽŽ . . The integer k NG C1, 3 , ᑜ, d, i is zero for all other ¨alues of d and i. Ž. Ž. Proof. Since CEG sZE, Proposition 1.2.2 says that the only 2-block of NEGŽ.is the principal block. By Proposition 1.2.3 it induces the principal 2-block ᑜ of G. By Proposition 2.1.4 we can replace NEGи2Ž.by E˜˜iS5 in counting the characters. The group E is an extra special group of order 25 . Bywx 10, IV.16.19 it has 16 linear representations and one irreducible representation with degree 4. Ž. The irreducible characters of NEG sE˜iA5 lying over the trivial character of E are those of A5, which have degrees 1, 3, 3, 4, and 5. The Ž. normalizer NEGи25sE˜iSinterchanges two of these characters with degree 3 and fixes the other three. Ä4 The commutator wxeeij,g is in "1 , for g g E˜ and 1 F i - j F 5. For each i, j with 1 F i - j F 5 we define a linear character ij on E˜sending any g g E˜to wxeeij,g. The resulting 10 characters form a single A5-orbit. The stabilizer in A51of the character 2in this orbit is the subgroup of A5 fixingÄ4 1, 2 and, hence, is isomorphic to S35. The stabilizer in S is the subgroup of S53fixingÄ4 1, 2 and, hence, is isomorphic to S = ޚ2. So lying over these 10 linear characters there are irreducible characters of E˜i A5 of degrees 10, 10, and 20. All of these characters are invariant under E˜i S5. Ä4 The commutator wxeeeijkl e,g is in "1 , for g g E˜ and 1 F i - j - k - lF5. For each i, j, k, l with 1 F i - j - k - l F 5 we define a linear character on E˜˜sending g g E to wxeeeijkk e,g. The resulting five linear characters form a single A5-orbit. The stabilizer of one of the characters in the orbit is A45in A and S 45in S . So lying over these five linear characters there are irreducible characters of E˜i A5 of degrees 5, 5, 5, and 15. Two of these characters with degree 5 are interchanged by E˜i S5. The other character of degree 5 and the one of degree 15 are fixed by E˜i S5. DADE’S CONJECTURE 601 The group E˜ has only one nonlinear character ⌳. The degree of ⌳ is 4. It is stabilized by E˜i A5. We claim that LEMMA 2.8.2. The Clifford extension for ⌳ in E˜i A5 is equi¨alent to 2 и A5. Proof. We first show that ⌳ cannot be extended to a character of the Sylow 2-subgroup E˜˜i V of E i A5, where V is the Sylow 2-subgroup Ä Ž.Ž.Ž.Ž.Ž.Ž.4 ²:² : 1, 12 34 , 13 24 , 14 23 of A51. The group F s ee2= eeee1234 (ޚ42=ޚis abelian and normal in E i V. Let be a linear character of ²: Ž.Ä4²Ž .2 : F with kernel eeee1234. Then is faithful on ZEs"1s ee12 . Ž . ² Ž .Ž . Ž .Ž .: Then E˜˜ s F and E i V s Fee1312 34 , 14 23 . If ⌳ can be ex- tended to E˜˜i V, then can be extended to Ž.E i V by Clifford theory Ž. Ä4 Ž. for F 1 E˜i V. Since ker s 1, eeee1234 we can see that Frker ( Ž.Ž. ޚ4 and F i V rker ( ޚ48( D , where ޚ 48( D is a central product. Ž. Let be the faithful linear character of Frker s ޚ4 obtained from . X Ä4 Linear characters of D88must be trivial on the derived group D s "1. So we cannot extend the faithful linear character of ޚ48( D . Therefore we cannot extend to Ž.F i V and, hence, we cannot extend to ⌳ to a character of E i V. Thus the Clifford extension for ⌳ in E˜i A5 does not split. Butwx 1, p. 2 tells us that the extension either is either split or equivalent to 2 и A55. So it must be equivalent to 2 и A , and Lemma 2.8.2 is proved. The degrees of the irreducible characters of NEGŽ.lying over ⌳ are 4 times the degrees for the irreducible characters of 2 и A5 nontrivial on ZŽ.2иA55, i.e., 8, 8, 16, and 24. The normalizer E i S interchanges the two characters of degree 8Ž seewx 1, p. 2. . This completes the proof of Lemma 2.8.1. 2.9. The Normalizer of the Radical 2-Chains C2, 2, C 2, 3, and C1, 4 The normalizer of each of the chains C2, 2, C 2, 3, and C1, 4 in G is ²: NTGŽ.sAB i Ž.Q = and in G и 2is ²: ²: ²: ²: NTGи2Ž.sAB i Ž.Q = i s AB i ŽQ = ., where is involution in Section 2.5. LEMMA 2.9.1. IfŽ.Ž.Ž.Ž. i, j s 2, 2 , 2, 3 , or 1, 4 , then there are 19 irre- ducible characters of NGiŽ. C ,ji. they all belong to the principal 2-block ᑜ ,j, which induces the principal 2-block ᑜ of G. Their degrees form the multiset 62623 Ä1,3,4,6,84 .The normalizer NGи2Ž. Ci, j fixes two characters of degree 1, two characters of degree 3, two characters of degree 4, two characters of degree 602 SONJA KOTLICA ŽŽ .. 6, and one character of degree 8. It mo¨es all other characters in Irr NCGi,j. The following holds: kNŽ.Ž.GiŽ. C,jGi,ᑜ,7,2 s4, kNŽ. C,j,ᑜ,7,2 s4 kNŽ.Ž.GiŽ. C,jGi,ᑜ,6,2 s2, kNŽ. C,j,ᑜ,6,2 s0 kNŽ.Ž.GiŽ. C,jGi,ᑜ,5,2 s2, kNŽ. C,j,ᑜ,5,2 s4 kNŽ.Ž.GiŽ. C,jGi,ᑜ,4,2 s1, kNŽ. C,j,ᑜ,4,2 s2. ŽŽ . . The integer k NGi C ,j, ᑜ, d, k is zero for all other ¨alues of d and k. Ž. Ž. Proof. Since CTG sZT, Proposition 1.2.2 says that each normal- izer NCGiŽ.,jihas only the principal 2-block ᑜ ,j. This block induces the principal 2-block ᑜ of G by Proposition 1.2.3. There are six linear characters of NTGŽ.lying over the trivial character on AB. The generator of Q is inverted under conjugation by , and is centralized by . So two of these characters are fixed by NTGи2Ž.and four are interchanged. The stabilizer of the six linear characters of AB trivial on one of ArŽ.Ž.AlBor Br A l B and not on the other is ²:²: 1 in Q = . Hence there is one irreducible character of NTGŽ.of degree 6 lying over these linear characters. Clearly this character is invariant under NTGи2Ž.. The nine linear characters of AB nontrivial on both ArŽ.A l B and BrŽ.AlBform two orbits under Q = ²: . One orbit has length 6. Any character in this orbit has stabilizer²: 1 in Q = ²: and some conjugate of ²: or ² :in ŽQ = ²:. i ²: . The other orbit has length 3. The stabilizer of any of its elements is ²: in Q = ²: and is conjugate to ²: = ²: in ŽQ = ²:. i ²: . Hence lying over the former orbit there is one irreducible character of NTGŽ.of degree 6. Lying over the latter orbit are two irreducible characters of degree 3. All of these characters are invariant under NTGи2Ž.. There are three nonlinear irreducible characters of AB, 0onAB y A l B ⌳ i s ½ 4i on A l B, where i, for i s 1, 2, 3, are the distinct nontrivial irreducible characters of A l B. The involution fixes one of the three characters i, say 1, and interchanges the other two. The stabilizer of ⌳1 in Q = ²: is Q = ²: and in ŽQ i ²:. = ²: it is Ž²:.²:Q i = . By Lemma 2.5.2 we can extend ⌳1 to an irreducible Ž. Ž²: ²:. character of NTGи21sAB Q = . So lying over ⌳ there are six irreducible characters of NTGŽ.of degree 4, two of them fixed and four interchanged by NTGи2Ž.. DADE’S CONJECTURE 603 The characters ⌳ 23and ⌳ are interchanged by . Their stabilizer in ²: Ž ²:. ²: ²: Q= is Q and in Q = i is isomorphic to Q i ( S3. Hence lying over them there are three irreducible characters of NTGŽ.of degree 8, two of which are interchanged by NT2иGŽ.. 2.10. The In¨ariant Conjecture p s 2 We use the results of the preceding seven sections to prove THEOREM 2.10.1. The In¨ariant Conjecture 1.3.4 holds for the prime p s 2. Proof. According to Proposition 2.3.2 there are eight conjugacy classes of radical 2-chains of G, represented by C0,1, C 1,1, C 1,2, C 2,1, C 1,3, C 2,2, C2, 3, and C 1, 4. By Lemma 2.4.1 there is only one block ᑜ in G with defect greater than 0. So Eq.Ž. 1.3.5 which we need to verify is kNŽ.GŽ. C0,1 ,ᑜ,d,iykNŽ.GŽ. C1,1 ,ᑜ,d,iykNŽ.GŽ. C1,2 ,ᑜ,d,i qkNŽ.GŽ. C2,1 ,ᑜ,d,iykNŽ.GŽ. C1,3 ,ᑜ,d,i qkNŽ.GŽ. C2,2 ,ᑜ,d,iqkNŽ.GŽ. C2,3 ,ᑜ,d,i ykNŽ.GŽ. C1,4 ,ᑜ,d,is0, Ž.) for any integer d G 0 and any i s 1, 2. By Lemmas 2.4.1, 2.5.1, 2.6.1, 2.7.1, 2.8.1, and 2.9.1 the left side of Ž.) is 4 y 4 y 0 q 0 y 4 q 4 q 4 y 4 s 0 for d s 7; i s 2, 2 y 2 y 4 q 4 y 2 q 2 q 2 y 2 s 0 for d s 6; i s 2, 2 y 2 y 0 q 0 y 2 q 2 q 2 y 2 s 0 for d s 5; i s 2, 0 y 0 y 1 q 1 y 1 q 1 q 1 y 1 s 0 for d s 4; i s 2, 1 y 0 y 0 q 0 y 1 q 0 q 0 y 0 s 0 for d s 3; i s 2, 4 y 4 y 0 q 0 y 4 q 4 q 4 y 4 s 0 for d s 7; i s 1, 4 y 4 y 12 q 12 y 0 q 0 q 0 y 0 s 0 for d s 6; i s 1, 0 y 4 y 0 q 0 y 0 q 4 q 4 y 4 s 0 for d s 5; i s 1, 0 y 0 y 2 q 2 y 2 q 2 q 2 y 2 s 0 for d s 4; i s 1. For all other values of d and i and all radical 2-chains C of G the number ŽŽ. . Ž. kNG C,ᑜ,d,i is zero. So, ) holds trivially for these d and i. PART III. THE PRIME p s 3 In this part we verify the Invariant Conjecture 1.3.4 for the prime p s 3. We start with a description of radical 3-chains of G. Then we analyze the normalizer of each chain and compute character degrees. 3.1. Radical 3-Subgroups and 3-Chains of G 604 SONJA KOTLICA Ž. Recall from Section 2.1 that the centralizer H s CzG of an involution z in G is the semi-direct product 0 H s E i A5 . 5 0 Here E is an extra special group of order 2 and A5 is a copy of the 0 alternating group A5. The group Q is the Sylow 3-subgroup²Ž 123 .: of A5. Bywx 12, Lemmas 3.2 and 5.2 the centralizer of Q in G has the form 0 CQGŽ.(Q=A6, Ž.3.1.1 0 where A66is the unique copy of A contained in CQGŽ.. We can choose an Ž. 0 Ž. involution 2 in NQG which inverts Q and acts on A622( PSL 9 as 2 in the notation of the Atlas wx1 . Then the normalizer of Q in G is 0 ²: NQGŽ.sŽ.Q=A62i (Ž.Q=A 62:2 (QiPGL 2Ž. 9 . A Sylow 3-subgroup W of CQGGŽ.is elementary of order 27 and NWŽ. Ž. lNQG sWiK, where K is a cyclic group of order 8 which acts Ž. Ž. faithfully on W and covers NQGGrCQ. So any generator of K inverts Q. Bywx 12, Lemma 5.4 the normalizer NWGŽ.has the form NWGŽ.sW1iK,Ž. 3.1.2 5 where W11is a Sylow 3-subgroup of G with order < WsQ=ZWŽ.1. Clearly, both factors Q and ZWŽ.1 in the decomposition are K-invariant. By lines 29 and 30 ofwx 12, p. 42 the group K acts fixed-point free on 2 ZWŽ.1 . The subgroup K of index 2 in K centralizes Q. It follows that Ž 2 .Ž. 2 Ž0. QsCKW and ZW136swW,K xgSyl A . Since K also operates fixed-point free on W1rW bywx 12, Lemma 5.4 , we have Q CKŽ.2. s W1 As a consequence of the properties described above we have the following lemma. DADE’S CONJECTURE 605 Ž. LEMMA 3.1.3. 1 1 ZW111W1WisaK-composition series for W1. Note that W1rW is irreducible under K because K acts regularly and Ž. transitively on the eight nontrivial elements of W11rW. Similarly, ZW is irreducible under K. PROPOSITION 3.1.4. The groups 1, W1, and Q are representati¨es for the distinct conjugacy classes of radical 3-subgroups in G. Proof. The above listed groups satisfy NGŽ.1 s G, OG3Ž.s1 NWGŽ.11sWiK, OW 31Ž.iKsW 1 NQGŽ.Ž.(Q=A62:2 , OQ 3Ž. Ž.=A 62:2 sQ. So they are radical 3-subgroups of G. Let X / 1 be a radical 3-subgroup of G. Then Y s ⍀ŽŽZX ..is a nontrivial elementary abelian 3-subgroup of Ž. Ž. Ž. Gwith NXGGFNY. Bywx 8, Lemma 3.4 , the normalizer NYGis G-conjugate to a subgroup of M, where M is either NQGGŽ.or NW Ž1 ..So Ž. we may replace X by some G-conjugate and assume that NXG FM. Then X is a radical 3-subgroup of M. Ž. Ž. Bywx 3, Proposition 1.4 , we have OM3 FX.If MsNWG 1 , then Ž. Ž.Ž . XsW13sOM.If MsNQGsQ=A62: 2 , then X s Q or X is NQGŽ.-conjugate to W, since the only nontrivial radical 3-subgroups of A6 are the Sylow 3-subgroups, which are trivial intersection subgroups of A6. Ž. But W is not a radical 3-subgroup of G because NWG sW1iK. PROPOSITION 3.1.5. There are exactly four conjugacy classes of radical 3-chains in G. These classes are represented by the following chains. For each chain we list its normalizer N in G, C0:1, NCGŽ.0sG 21q2 C11:1-W , NCGŽ.11sWiK(3и3:8 C1,0 :1-Q, NCGŽ.1,0. (Ž.Q=A62:2 C2:1-Q-W, NCGŽ.21sŽ.Q=ZW Ž.iK. Proof. Clearly, C0 is the only radical 3-chain in G of length zero. If a radical 3-chain in G has length 1, it is 1 - W1 or 1 - Q. If it has length 2, it must start with 1 - Q followed by a radical 3-subgroup X of NQGŽ.. 0 Any such X has the form Q = Y, where Y is a radical 3-subgroup of A6. The only radical 3-subgroups of A6 are 1 and the Sylow 3-subgroups. Ž. Ž. Hence we may assume that Y s ZW11and X s Q = ZW sW. Since Ž . Ž . Ž Ž .. ŽŽ Ž .. . NQGGlNWsQ=ZW131iK and OQ=ZW iK sW, the chain 1 - Q - W is a radical 3-chain of G. 606 SONJA KOTLICA 3.2. The Character Degrees of the Normalizer of C0 LEMMA 3.2.1. The degrees of the irreducible characters of G form the multiset Ä1, 8522 , 323 , 324, 646 2 , 816, 1140, 12152 , 1615, 19203 , 1938 2 , 2432, 2754, 30784 . The irreducible characters of G with degrees 324, 2754, and 3078 belong to a single 3-block ᑜ1 with defect 1. The two irreducible characters with degree 1215 belong to two separate 3-blocks with defect 0. All other irreducible characters belong to the principal 3-block ᑜ 0. The group G и 2 interchanges the two irreducible characters of degree 85, the two of degree 323, the two of degree 646, and the two characters of degree 1938. It fixes all other irreducible characters of G. Hence we ha¨e following table: kGŽ.,ᑜ00,5,2 s3, kG Ž.,ᑜ,5,1 s6 kGŽ.,ᑜ00,4,2 s5, kG Ž.,ᑜ,4,1 s2 kGŽ.,ᑜ11,1,2 s3, kG Ž.,ᑜ,1,1 s0. Ž. The integer k G, ᑜ k , d, i is zero for all other ¨alues of k, d, and i. Proof. Here, as in Proposition 2.4.1, we make use of the Proposition 1.2.4. In the Atlas wx1 620, , and 21denote the irreducible characters of Gwith degrees 324, 2754, and 3078. Using the formula in Proposition 1.2.4 we determine that they belong to the same 3-block ᑜ1 and that no other characters belong to that block. The other 3-blocks are determined simi- larly. 3.3. The Character Degrees of the Normalizer of C1 The normalizer in G of the chain C11:1-W is 2 1q2 NWGŽ.11sWiKs3и3 :8. We can see fromwx 1, p. 82 that in G и 2 the normalizer of C1 has the form NWGи21Ž.sW 1iD, where D is nonabelian group of order 16 and contains K. The group D normalizes K, K 2, and W . Hence D normalizes Q CKŽ 2 .. 1 s W1 ࠻ Since D permutes the eight elements of ZWŽ.1 , while the subgroup K is regular and transitive on these eight elements, the stabilizer Dw of some Ž.࠻ element w g ZW1 satisfies ²: LEMMA 3.3.1. Dw s has order 2, is complementary to K in D s K ²: Ž. i,and centralizes the nontri¨ial element w of Z W1 . DADE’S CONJECTURE 607 We next prove LEMMA 3.3.2. The group D acts faithfully on ZŽ. W1 by conjugation in Gи2. Hence D s K i ²: is a semi-dihedral group of order 16. Proof. If CZWDŽ Ž1 ../1, then CZWD Ž Ž1 ..must be ² :, i.e., central- Ž. izes ZW1 . The involution g G и 2 y G must belong to the class 2B in the notation of the Atlas wx1, p. 83 . The Atlas tells us that the centralizer in 42 Ž. Gof an involution of class 2B has order 2448 s 2 и 3 и 17. Hence ZW1 Ž. Ž . Ž. must be a Sylow 3-subgroup of CG . In particular, ZW1 sCW .It Ž.1 follows that the involution acts without fixed points on W11rZW . This Ž. Ž . forces W11rZW to be abelian seewx 10, V.18.8 . This is false since Ž. W11rZW is nonabelian by the last line ofwx 12, Lemma 5.4 . Hence ŽŽ .. Ž. CZWD 11s1 and D acts faithfully on ZW . Because a Sylow 2-sub- ŽŽ .. Ž. group of Aut ZW12(GL 3 is semi-dihedral group of the same order 16 as D, this forces D to be a semi-dihedral group K i ²: of order 16. LEMMA 3.3.3. The centralizer CDŽ. Q is the quaternion subgroup of order 8 Ž. in D. Any element of D y CQinD ¨erts Q. In particular, in¨erts Q. Ž. Proof. By Lemma 3.1.3 the group X s W11rZW has a K-composition Ž. series X ) Y s WrZW1 )1 with elementary abelian factor groups XrY 2 ( W1rW and Yr1 ( Q of orders 3 and 3, respectively. It follows that X is either elementary abelian of order 33 or extra-special of order 31q2 and exponent 3. But X is nonabelian by the last line ofwx 12, Lemma 5.4 . Hence 1q2 Ž. Ž . Xis extra special of order 3 and exponent 3. Then ZX sWrZW1 Ž. ŽŽ .. (Qand CQDDsCZX. The conjugation action of the semi-dihedral Ž. Ž. group D on XrZX (W1rWis faithful because ZD is a subgroup of K and K acts faithfully. Ž. Ž Ž.. Ž. Observe that Aut X is isomorphic to XrZX iGL2 3 . The sub- group of AutŽ.X centralizing ZXŽ.is sent by this same isomorphism onto Ž Ž .. Ž. Ž. XrZX iSL22 3 . The Sylow 2-subgroup of SL 3 is well known to be a quaternion group of order 8. LEMMA 3.3.4. All irreducible characters of NGŽ. C1 belong to the principal 3-block ᑿ 00which induces the principal 3-block ᑜ of G. Their degrees form 84 3 the multiset Ä1 , 6 , 8, 244 . The normalizer NGи21Ž. C fixes two linear charac- ters of NGŽ. C1 , two irreducible characters of degree 6, one irreducible charac- ter of degree 8, and all irreducible characters of degree 24. It mo¨es all the remaining irreducible characters of NGŽ. C1 . Hence the following holds: kNŽ.Ž.GŽ. C10,ᑜ,5,2 s3, kNGŽ. C10,ᑜ,5,1 s6 kNŽ.Ž.GŽ. C10,ᑜ,4,2 s5, kNGŽ. C10,ᑜ,4,1 s2. ŽŽ. . The integer k NG C1 , ᑜ k , d, i is zero for all other ¨alues of k, d, and i. 608 SONJA KOTLICA Proof. The centralizer CWGŽ.11is the center ZW Ž.. So by Proposition 1.2.2 all irreducible characters of NCGŽ.1 belong to the principal 3-block ᑿ 00, which induces the principal 3-block ᑜ of G by Proposition 1.2.3. Over the trivial character of W1 there are eight linear characters of Ž. 3 NCG 11sWiK. The element acts on K as ª . So there are two linear characters of NWGŽ.1 which are -invariant and six which are moved by . The nontrivial irreducible characters of W1 trivial on W are all linear and form a single orbit of length 8 under the action of K. Lying over this orbit there is exactly one character of NWGŽ.1 with degree 8. This charac- ter is invariant under NWGи21Ž.. Ž. 1q2 Ž. W11rZW is extra-special of order 3 with center WrZW1(Q.So it has one irreducible character of degree 3 lying over each nontrivial Ž. linear character of WrZW11. It follows that W has one irreducible character ⌳ of degree 3 lying over each nontrivial linear character of W trivial on ZWŽ.1 . There are exactly two such , interchanged by any generator of K. So they form a single D-orbit. Because any such is faithful on Q and trivial on ZWŽ.1 , its stabilizer D is the quaternion subgroup CQDŽ.in Lemma 3.3.2. This must also be a stabilizer D⌳ of the Ž . 3 unique character ⌳ g Irr W1 < . Because < f:Q=W11ªZWŽ., given by fqŽ.,wW11swxq,w for any q g Q and w11g W ,is D-invariant and bilinear. If q g Q is not 1, ²: Ž. then q s Q and wxq, w11/ 1 for all w g W1y CQWsW1yW. Since Ž. 1Ž. < Ä Ž. Ž.4 This isomorphism must send W1, s w11g W < fQ,w 1;ker 1onto kerŽ.1 .So f restricts to a nonsingular bilinear map of Q = W1, into Ž. Ž.²: ker 11s wZW , x. Since the involution in D inverts both Q and kerŽ.11it must centralize W ,. Hence lying over there are three linear characters of W1, , each of which is invariant under . Inducing these linear characters to NWGŽ.1 we obtain the three irreducible characters of NWGŽ.1 of degree 24 lying over the orbit of . Furthermore, each of these Ž. Ž. ²: three characters is fixed by NWGи21sNWG1i. 3.4. The Character Degrees of the Normalizer of C1, 0 The normalizer in G of C1, 0 :1-Q is NQGŽ.( ŽQ=A62 .:2 . We can see inwx 1 that in G и 2 the normalizer is NQGи2632102Ž.( ŽQ=Aи2:2 .s ŽQ=M .:2 . LEMMA 3.4.1. There are 18 irreducible characters of NGŽ. C1, 0 . Their degrees form a multiset Ä124252 , 2, 8 , 9 , 10 , 16 , 18, 204 . All these characters belong to the principal 3-block ᑿ 0 except the two characters of degree 9 and the character of degree 18, which belong to a different block ᑿ11. The block ᑿ induces the block ᑜ10, and the block ᑿ induces the block ᑜ 0. The normal- izer NGи21,0Ž. C fixes two linear characters of NGŽ. C1, 0 , one irreducible charac- ter of degree 2, two irreducible characters of degree 9, one of degree 10, one of degree 18, and one of degree 20. It mo¨es all other irreducible characters of NCGŽ.1, 0 .Hence the following holds: kNŽ.Ž.GŽ. C1,0,ᑜ 0 ,3,2 s5, kNGŽ. C1,0,ᑜ 0 ,3,1 s10 kNŽ.Ž.GŽ. C1,0,ᑜ 1,1,2 s3, kNGŽ. C1,0,ᑜ 1,1,1 s0. ŽŽ . . The integer k NG C1, 0 , ᑜ k, d, i is zero for all other ¨alues of k, d, and i. Proof. From the character table A66inwx 1 we see that A has two 3-blocks, a 3-block ᑜ16Ž.A of defect 0, containing the irreducible charac- ter 60of degree 9, and the principal 3-block ᑜ Ž.A6containing all other Ž. irreducible characters of A6. It follows that QCG Q s Q = A6 has two 3-blocks, the 3-block ᑜ16Ž.Q = A , lying over ᑜ16 Ž.A and having Q as a defect group, and the principal 3-block ᑜ 06Ž.Q = A . Each of these last Ž. Ž . Ž. Ž. two blocks is invariant in NQ s Q=A62: 2 . Since wNQ:QC Q xs 2 is not divisible by 3, we conclude that NQŽ.has two 3-blocks, the 3-block ᑿ11, lying over ᑜ Ž.Q = A6and having Q as a defect group, and the principal 3-block ᑿ 00, lying over ᑜ Ž.Q = A6. The principal block ᑿ 0 610 SONJA KOTLICA induces the principal block ᑜ 0 by Proposition 1.2.3. By Brauer’s first main GG theoremwx 7, 9.7 , the induced block ᑿ11has Q as a defect group. So ᑿ must be ᑜ1. 00 The centralizer CQGи21010Ž.has the form Q = M , where M is a copy of 0 M10sA 6и2 3 containing A6. The degrees of the irreducible characters of NCGŽ.1, 0 which lie over nontrivial characters of Q are two times the degrees of the irreducible characters of A6 : 2, 10, 10, 16, 16, 18, 20wx 1, p. 5 . The two characters of degree 10 are interchanged by NQGи2Ž.as are the two characters of degree 16wx 1, p. 5 . The other characters are fixed by NQGи2Ž.. The degrees of the irreducible characters of NQGŽ.which lie over the trivial character of Q are the degrees of irreducible characters of A62и 2, namely 1, 1, 10, 8, 8, 8, 8, 9, 9, 10, and 10wx 1, p. 5 . The four characters of degree 8 are all moved by NQGи2Ž.wx1, p. 5 . The two characters of degree 1 remain fixed under the action of NQGи2Ž.because one of them is the trivial character. The irreducible character of degree 9 of A66can be extended to AutŽ.A Ž. Ž since its degree is relatively prime to the indexw Aut A66: A xs 4 seew 11, Theorem 8.15x. . So both of the characters of A62и 2 of degree 9 can be extended to irreducible characters of NQGи2Ž.. Therefore they remain fixed under the action of NQGи2Ž.. The following argument shows that the two extensions to A62и 2 of the irreducible character 76of A with degree 10Ž in the notation ofwx 1, p. 5 . are moved by NQGи2Ž.. There are two classes of elements of order 8 in PGLŽ. 9 NQ Ž . Q. The element x 0 , where is a generator of 2 ( G r s ž/01 the multiplicative group of ކ92, has order 8 in GLŽ. 9 and its image x has order 8 in PGL22Ž. 9 . One noninner automorphism of PGLŽ. 9 is the Galois automorphism, which sends x to x 33. If the elements x and x were Ž. Ž. conjugate in PGL22 9 , there would exist an element y g GL 9 such that 31i xsyxyyŽI.for some i. This cannot happen since the eigenvalues of x33are and 1 and the eigenvalues of yxyy1ŽiI.are iq1and i. Hence the Galois automorphism of PGL2Ž. 9 exchanges the two classes of elements of order 8 in that group. But the two extensions of 7 to PGL2Ž. 9 of degree 10 have different values on those two classesŽ the classes 8A and 8B inwx 1, p. 5. . 3.5. The Character Degrees of the Normalizer of C2 The normalizer in G of the chain C2 :1-Q-W is Ž.Q = ZWŽ.1iK. The normalizer of C2 in G и 2is Ž.Q=ZWŽ.1iD. DADE’S CONJECTURE 611 LEMMA 3.5.1. All irreducible characters of NGŽ. C2 belong to the principal 3-block, which induces the principal 3-block ᑜ 0 of G. Their degrees form the 843 multiset Ä1,2,84 . The normalizer NGи22Ž. C fixes two linear characters of NCGŽ.2 ,two irreducible characters of degree 2 and one irreducible character Ž. of degree 8. It mo¨es all other irreducible characters of NG C2 . Hence, the following holds: kNŽ.Ž.GŽ. C20,ᑜ,3,2 s5, kNGŽ. C20,ᑜ,3,1 s10. ŽŽ. . The integer k NG C2 , ᑜ k , d, i is zero for all other ¨alues of k, d, and i. ŽŽ..Ž. Proof. Since CQG =ZW11sQ=ZW all irreducible characters of NCGŽ.2 belong to the principal 3-block by Proposition 1.2.2. This block induces the principal 3-block ᑜ 0 of G by Proposition 1.2.3. Over the trivial character of Q = ZWŽ.1 there are 8 linear characters of NCGŽ.2. Two of these are invariant and six moved under NCGи22Ž.. The stabilizer in NCGŽ.2 of each linear character of W trivial on 2 ZWŽ.1 and nontrivial on Q is W i K . The stabilizer of in NCGи22Ž.is WiCQDDŽ., where CQ Ž.is the quaternion group in Lemma 3.3.2. So lying over this orbit there are four irreducible characters of NCGŽ.2 with degree 2, of which two are invariant under NCGи22Ž.and two are not. There are 24 linear characters of W nontrivial on ZWŽ.1 . They form three regular orbits under the action of K. Each of these 24 characters Ž. has stabilizer W in NCG 2 sWiK. So it induces the unique irreducible character W i K of W i K lying over the conjugacy class of . This character W i K has degree 8. Hence lying over each of the three orbits is one irreducible character of NCGŽ.2 of degree 8. Ž. Let 11be the nontrivial character on ZW fixed by .If s1Q=1, then the stabilizer of is W in W i K and W i ²: in W i D. So, the character belongs to a W i D-orbit of length 8 and W i K is an irreducible character of degree 8 fixed by W i D. Now suppose that s = 1, where is a nontrivial character of Q. We claim that the D-orbit of has length 16. If not, then the stabilizer D is a subgroup ² X : of order 2 in D complementary to K. But then X fixes and centralizes Q. This is impossible, since the only involution in the quaternion group CQDŽ.belongs to K. Hence belongs to a W i D-orbit of length 16. We conclude that W i K induces an irreducible character W i D of Ž. Ž. WiDand, hence, is not fixed by W i D s NCGи22. Thus NCGи22fixes one irreducible character of NCGŽ.2 with degree 8 and interchanges the other two such characters. 3.6. The In¨ariant Conjecture for p s 3 The results of the preceding five sections can now be used to prove THEOREM 3.6.1. In¨ariant Conjecture 1.3.4 holds for the prime p s 3. 612 SONJA KOTLICA Proof. According to Proposition 3.1.5 there are four conjugacy classes of radical 3-chains, represented by C011,0, C , C , and C 2. So Eq.Ž. 1.3.5 which we need to verify is kNŽ.GŽ. C0,ᑜkG,d,iykNŽ.Ž. C1,ᑜk,d,i ykNŽ.GŽ. C1,0 ,ᑜkG,d,iqkNŽ.Ž. C2,ᑜk,d,is0. Ž.) for any k s 0, 1, any integer d G 0, and any i s 1, 2. By Lemmas 3.2.1, 3.3.4, 3.4.1, and 3.5.1, the left side of this equation is 3 y 3 y 0 q 0 s 0 for k s 0; d s 5; i s 2, 6 y 6 y 0 q 0 s 0 for k s 0; d s 5; i s 1, 5 y 5 y 0 q 0 s 0 for k s 0; d s 4; i s 2, 2 y 2 y 0 q 0 s 0 for k s 0; d s 4; i s 1, 0 y 0 y 5 q 5 s 0 for k s 0; d s 3; i s 2, 0 y 0 y 10 q 10 s 0 for k s 0; d s 3; i s 1, 3 y 0 y 3 q 0 s 0 for k s 1; d s 1; i s 2. For all other values of k, d, and i and all radical 3-chains C in G the ŽŽ. . Ž. number kNGk C,ᑜ,d,i is zero. So ) holds trivially in these cases. PART IV. THE GROUP 3 и G In this part we verify the projective conjecture 1.4.2 for 3 и G s 3 и J3 and the primes p s 3 and p s 2. We start with a description of Sylow U 3-subgroup W13of 3 и J . In Section 4.2 we compute the character degrees of the normalizers in 3 и J33of radical 3-chains of J . The above conjecture is verified for p s 3 in Section 4.3 and for p s 2 in Section 4.4. U 4.1. The Structure of W1 If M is a subgroup of G, let MU be the corresponding subgroup 3 и M U Ž. of 3 и G containing Z3 s Z 3 и G . U There is a well-defined conjugation action of G on 3 и G s G with any U U g g G acting as conjugation by any g g G having g as its image in G. The cyclic group K of order eight in Section 3.1 normalizes W1, W, Q, and UUU U ZWŽ.111. So, it normalizes W , W , Q , and ZWŽ.. The group K acts U regularly and transitively on the nontrivial elements of W1 rW and on Ž.UU U those of ZW13rZ . It inverts elements of QrZ3. Since 11ZWŽ.111W1W DADE’S CONJECTURE 613 is a K-composition series for W Ž.see Lemma 3.1.3 , one K-composition U series for W1 is UUUU 11Z311ZWŽ.1W 1W 1. We shall examine the groups in this series. We need the following proposition. PROPOSITION 4.1.1. No faithful simple ޚ3Ž.K -module V is isomorphic to its dual ޚ Ž.K -module Vˆ Hom Ž.V , ޚ . 3 s ޚ 3 3 ࠻ Proof. Since the multiplicative group ކ99of the field ކ of order 9 is cyclic of order 8 and, hence, isomorphic to K, there is some ޚ3-isomor- phism of V onto ކ93as ޚ -modules carrying multiplication by any ࠻ generator of K into multiplication by some generator of ކ9 . It follows that multiplication by is a ޚ3-linear transformation T of V with 3 eigenvalues and in ކ9. Multiplication by is the inverse dual transformation Tˆy1 of Vˆ with eigenvalues y1 and y3. Since neither of y1 y33 or is equal to or , it follows that V is not ޚ3Ž.K -isomor- phic to Vˆ. U PROPOSITION 4.1.2. The group ZŽ. W1 is elementary abelian. In fact UUUU ZWŽ.13sZ = ZW Ž. 1,K (Z 3=ZWŽ. 1. Proof. By Lemma 4.1.1 the irreducible representation of ޚ3Ž.K on q ZWŽ.1 induced by conjugation in G is not self-dual. It follows that the commutator map UUUUU ZWŽ.13rZ =ZW Ž. 13rZ ªZ 3 UU UU UU Z33=Zªw,x U is trivial. Hence ZWŽ.1 is abelian. Bywx 10, III.13.4 if a group X acts on Ž . Ž. abelian group Y and <<< UU U U WsZ33=wW,Kx(Z=W. Ž.U Ž.U 2 Ž. 2 Ž. Proof. Note that wZW11,Kxws ZW ,K xw( ZW 11,K xsZW UŽ.U by Proposition 4.1.2. A generator of the group K inverts W rZW1 ( U 42 Qand has order 8 relatively prime to U Ž.U of order 4 in K centralizes W rZW1 and, hence, centralizes some UU U 2 Ž.Ž .ŽU. gWyZW1 seewx 10, I.18.6 . The subgroup CKW has order 2 U UŽ. 3 . Its subgroup CKW sZ3 has order 3. The group K inverts 2 UU2 UŽ . UŽ. CKW rZ33and centralizes Z . Hence CKWcannot be cyclic and its U U element has order 3. Commutation in W induces a ޚ3-bilinear map U q ²:UUq q f:ZWŽ.13,K = ªŽ.Z 2U q that preserves the action of K . Since wZWŽ.1 ,Kxis an irreducible 2 Uq 2 ޚ3Ž K.²-module of dimension 2 and :is an irreducible ޚ3ŽK .-mod- 2 ule of dimension 1, their tensor product is an irreducible ޚ3Ž K .-module of dimension 2. Hence f is trivial. It follows that W U is elementary abelian and satisfies UU U U U WsZ333=wW,Kx(Z=wxW,KsZ=W. U PROPOSITION 4.1.4. Commutation in W1 induces a nondegenerate K- UU Ž.UU U in¨ariant bilinear map W11rW = ZW rZ3ªZ3. Proof. The group W U is abelian by Proposition 4.1.3. So commutation U in W1 induces a K-invariant bilinear map UUqqU Uq U e:Ž.Ž.Ž.W11rW=Ž.ZW rZ3ª Z3. ŽUU.q ŽŽ .UU.q Since W11rW and ZW rZ3are both simple ޚ 3wxK -modules, e U is either trivial or nondegenerate. Suppose it is trivial. Then wZWŽ.1 ,Kxis UUU a normal subgroup of W13, complementary to Z in ZWŽ.1. We use bars UU U to denote images modulo this normal subgroup. Then W s Z3 = wxW , K . UU Note that wxW , K s ²: ( Q. U Commutation in W13now induces a K-invariant, ޚ -bilinear map UUqqq UU U f:Ž.Ž.Ž.W13rW=WrZªZ3. Ž.UUq Ž. Since W1 r W is a two-dimensional irreducible ޚ3 K -module, Ž.UUq Ž. WrZ33is an irreducible one-dimensional ޚ K -module, and their tensor product is a two-dimensional irreducible ޚ3Ž.K -module, it follows UU that f has to be trivial. Hence the subgroup wxW , K is normal in W1 . We use another bar to denote images modulo the normal subgroup UU UUU wxW,Kof W13. We get W s Z ( Z3. Commutation in W 1induces a K-invariant bilinear map UUUqqq g:Ž.Ž.Ž.W13rZ=W 13rZªZ 3. Ž. ŽU .qq Ž . Ž But the simple ޚ31K -module W r Z31( W rW is not self-dual see U Lemma 4.1.1. . Therefore g is trivial and W1 is abelian. DADE’S CONJECTURE 615 UU U This implies that W13s Z = wxwW 1, K by 10, III.13.4 x . So the inverse UUU image of wxW13, K is a complement to ޚ in W11. That is not true since W U is a Sylow 3-subgroup of G and Z3 does not have a complement in 3 и G. U U So we must have a nondegenerate commutator map W1 rW = Ž.UU U ZW13rZ ªZ 3. 4.2. The Radical 3-Chains and Their Normalizers PROPOSITION 4.2.1. There are exactly four conjugacy classes of radical 3-chains in G. These classes of radical 3-chains are represented by the following chains. For each chain we list its normalizer NU in GU. U U C0 :1, N sG UU C11:1-W , N (W18iޚ UU C1,0 :1-Q, N (ޚ3 =Ž.Ž.Q=A6:2 2(Z 3 =NCG Ž.1,0 UU C23:1-Q-W, N (ޚ =Ž.Ž.Q=ZWŽ.1iޚ8(Z3=NCGŽ.2. Proof. That there are exactly four conjugacy classes of radical 3-chains follows from Proposition 3.1.5. The normalizers of C1, 0and C 2 split over U U Z3 because their common Sylow 3-subgroup W splits by Proposition 4.1.3. LEMMA 4.2.2. There are 17 irreducible characters of 3 и G lying o¨er o¨er U a fixed nontri¨ial linear character 33of Z . Their degrees form a multiset Ä18223 , 153 , 171 , 324, 1215 2 , 1530 2 , 2736, 2754, 2907, 3060, 30784 . The charac- ters with degrees 18, 18, 153, 153, 171, 171, 171, 1530, 1530, 2736, 2907, and U 3060 belong to the principal 3-block ᑜ 0 of 3 и G with defect 6. The U characters with degrees 324, 2754, and 3078 belong to a different 3-block ᑜ1 with defect 2. The two characters with degree 1215 each belong to a separate U 3-block with Z3 as a defect group. The following holds: U kŽ.3 и G, ᑜ 03,4< s12 U kŽ.3иG,ᑜ13,2< s3. ŽU . The integer k 3 и G, ᑜ i , d<3 is zero for all other ¨alues of i and d. Proof. Directly fromwx 1, p. 83 and the usual formula for characters in blocks. LEMMA 4.2.3. The normalizer N3иGŽ. C11 of the chain C has 12 irreducible U characters lying o¨er a gi¨en nontri¨ial linear character 33of Z . Their degrees form the multiset Ä984 , 184 . All these characters belong to the principal U U 3-block ᑿ13of NиGŽ. C10 which induces the block ᑜ . The following holds: U kNŽ.3иGŽ. C10,ᑜ,4< 3s12. ŽŽ. . The integer k N3иG C1 , ᑜ i, d<3 is zero for all other ¨alues of i and d. 616 SONJA KOTLICA Ž U .ŽU. Proof. Since CWG 11sZW , all characters belong to the principal UU block by Proposition 1.2.2. The subgroup ZWŽ.11is normal in W i K.So we can apply Clifford theory to this normal subgroup. The nontrivial linear U U character 33of Z has nine extensions to the abelian group ZWŽ.1.It U follows from Proposition 4.1.4 that the group W1 permutes these exten- sions transitively with W U as the stabilizer of each extension. One of them, say U, is also fixed by K. Hence U has W U i K as its stabilizer in UU U U NC3иGŽ.1 . One extension of to W is fixed by K. Thus extends to eight linear characters of W U i K which induce eight characters of U U U U U degree 9 in W11i K. The other two extensions , 2of to W are 2 U U 2 interchanged by any element of K y K . The stabilizer of 1 is W i K . U The four linear extensions of 1 to this stabilizer induce four irreducible U characters of W1 i K, each of degree 18. U Since the normalizers of C1, 0 and C2 split over Z3 the following two lemmas follow directly from Lemmas 3.4.1 and 3.5.1. Ž. LEMMA 4.2.4. There are 18 irreducible characters of N3иG C1, 0 lying o¨er U a fixed nontri¨ial character 33of Z . Their degrees form the multiset Ä124252 , 2, 8 , 9 , 10 , 16 , 18, 204 . All these characters belong to the principal 0U U block ᑿ1, 0 , which induces ᑜ 0 , except the two characters of degree 9 and the 1U U character of degree 18, which belong to a block ᑿ1, 0 inducing ᑜ1 . So U kNŽ.3иGŽ. C1,0,ᑜ 0,4< 3 s15 U kNŽ.3иGŽ. C1,0,ᑜ 1,2< 3 s3. ŽŽ.U . The integer k N3иG C1, 0 , ᑜ i , d<3 is zero for all other ¨alues of i and d. 0U Proof. The principal 3-block ᑿ1, 0of NC 3иGŽ.1, 0 induces the principal block ᑜ 0 of 3 и G by Proposition 1.2.3. It follows from Lemma 3.4.1 that 1U Ž. UU the other 3-block ᑿ1, 0of NC 3иG 1, 0has Q ( Z 3 = Q as a defect group. Ž. ŽU.Ž. Since NC3иG 1, 0sNQ 3иG as a defect group. Since NC3иG 1, 0 s U NQ3иGŽ ., Brauer’s first main theorem on blockswx 7, III.9.7 implies that 1U U ᑿ1, 0 induces a 3-block of 3 и G having Q as a defect group. This 3-block can only be ᑜ1. Ž. LEMMA 4.2.5. There are 15 irreducible characters of N3иG C2 lying o¨er a U fixed nontri¨ial character 33of Z . Their degrees form the multiset ŽŽ.Ä8434 deg NC3иG 2 s1,2,8 . All these characters belong to the principal 3- U U block ᑿ 20which induces the principal 3-block ᑜ . So U kNŽ.GŽ. C20,ᑜ,4< 3s15. ŽŽ.U . The integer k NG C2 , ᑜ i , d<3 is zero for all other ¨alues of i and d. DADE’S CONJECTURE 617 Proof. This follows from Lemma 3.5.1. 4.3. The Projecti¨e Conjecture for 3 и G and p s 3 The results of the previous section will give us THEOREM 4.3.1. The projecti¨e conjecture 1.4.2 holds for Z и G s 3 и G s 3 и J3 and the prime p s 3. U Proof. We fix a faithful irreducible character 33s of Z s Z. Since U Z3 is its own Sylow 3-subgroup, it follows from Lemma 4.2.2. that the only U U blocks ᑜ of 3 и G not having Z30as a defect group are ᑜ and ᑜ1. According to Proposition 4.2.1 there are four conjugacy classes of radical 3-chains represented by C011,0, C , C , and C 2. Hence Eq.Ž. 1.4.3 which we must verify is U U kNŽ.3иGŽ. C0,ᑜi,d< 33ykNŽ.иGŽ. C1,ᑜi,d< 3 U U ykNŽ.3иGŽ. C1,1 ,ᑜi ,d<33qkNŽ.иGŽ. C2,ᑜi,d<3s0 Ž.) for any d ) 0 and i s 1, 2. By Lemmas 4.2.2, 4.2.3, 4,2.4, and 4.2.5 the left side of Ž.) for i s 1 and d s 4is 12 y 12 y 15 q 15 s 0. The same lemmas imply that the left side of Ž.) for i s 2 and d s 2is 3y0y3q0s0. For any other values of i and d the left side of Ž.) is 0 y 0 y 0 q 0 s 0. Hence Ž.) holds in all cases. 4.4. The Projecti¨e Conjecture for 3 и G and p s 2 Fromwx 1 we know that: LEMMA 4.4.1. There are 17 irreducible characters of 3 и G lying o¨er a U fixed nontri¨ial linear character 33of Z . They all belong to a single 2-block ᑜUof 3 и G. Their degrees form the multiset Ä1822 , 153 , 171 3 , 324, 1215 2 , 15302 , 2736, 2754, 2907, 3060, 30784 . The following holds: U kŽ.3 и G, ᑜ ,7< 3s8 U kŽ.3иG,ᑜ,6<3s6 U kŽ.3иG,ᑜ,5<3s2 U kŽ.3иG,ᑜ,4<3s0 U kŽ.3иG,ᑜ,3<3s1. ŽU. The integer k 3 и G, ᑜ , d<3 is zero for all other ¨alues of d. 618 SONJA KOTLICA Proof. Directly fromwx 1, p. 83 . THEOREM 4.4.2. The projecti¨e conjecture 1.4.2 holds for Z и G s 3 и G s 3 и J3 and the prime p s 2. Proof. If C is a radical 3-chain of G with length ) 0, then it follows from Proposition 2.3.2 that a 3-Sylow subgroup Y of NCGŽ.is elementary abelian. By the proof ofwx 12, Lemma 5.8 this implies that Y is G-conjugate to a subgroup of W. We know from Proposition 4.1.3 that W U splits over Ž. Ž.UU ޚ33. Hence NCиGGsNC splits over Z3. So, the degrees of irre- ducible characters of NC3иGŽ.lying over a fixed nontrivial character of ޚ3 are the same as the degrees of the irreducible characters of NCGŽ.. According to Proposition 2.3.2 there are eight conjugacy classes of radical 2-chains in G, represented by C0,1, C 1,1, C 1,2, C 2,1, C 1,3, C 2,2, C 2,3, and C1, 4. So Eq.Ž. 1.4.3 which we need to verify is U U kNŽ.3иGŽ. C0,1,ᑜ,d< 3ykNŽ. 3иGŽ. C1,1,ᑜ,d< 3 U U ykNŽ.3иGŽ. C1,2,ᑜ,d< 3qkNŽ. 3иGŽ. C2,1,ᑜ,d< 3 U U ykNŽ.3иGŽ. C1,3,ᑜ,d< 3qkNŽ. 3иGŽ. C2,2,ᑜ,d< 3 U U qkNŽ.3иGŽ. C2,3,ᑜ,d< 3ykNŽ. 3иGŽ. C1,4,ᑜ,d< 3 s0, Ž.) for any d ) 0. By Lemmas 4.4.1, 2.5.1, 2.6.1, 2.7.1., 2.8.1, and 2.9.1 the left side of Ž.) is 8 y 8 y 0 q 0 y 8 q 8 q 8 y 8 s 0 for d s 7; 6 y 6 y 16 q 16 y 2 q 2 q 2 y 2 s 0 for d s 6; 2 y 6 y 0 q 0 y 2 q 6 q 6 y 6 s 0 for d s 5; 0 y 0 y 3 q 3 y 3 q 3 q 3 y 3 s 0 for d s 4; 1 y 0 y 0 q 0 y 1 q 0 q 0 y 0 s 0 for d s 3. For all other values of d and all radical 2-chains C of G the number ŽŽ.U .Ž. kN3иG C,ᑜ,d<3 is zero. Hence ) holds for all d, and the theorem is proved. REFERENCES 1. J. Conway, R. Curtis, S. Norton, R. Parker, and R. 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