Ž .JOURNAL OF ALGEBRA 187, 203]213 1997ARTICLE NO. JA976798

New Properties of the p-Special Characters

Gabriel Navarro*, †

Departament d’Algebra, Facultat de Matematiques, Uni ersitat de Valencia,46100 Burjassot, Valencia, Spain

Communicated by George Glauherman

Received February 14, 1996

I. INTRODUCTION

Since its discovery by D. Gajendragadkar in 1978, the p-special charac-ters have played a prominent role in the development of the CharacterTheory of Solvable Groups. One of the key reasons for that, certainlyamong others, is the fact that every irreducible character in a finitesolvable group is induced from a ‘‘p-factorable’’ character, i.e., a characterwhich is a product of a p-special times a p 9-special character.

In character theory it is important to know as much as possible abouthow we may construct new irreducible characters from old ones. It is,sometimes, even better if the process involves induction of characters sincethis usually provides interesting interactions between the group and cer-tain subgroups.

The aim of this paper is to prove the following.

THEOREM A. Suppose that G is sol able of odd order. Let H be aŽ . Gsubgroup of G and let a g Irr H be a p-special character such that a is

Ž . Ž .Girreducible. If b g Irr H is p 9-special, then ab is irreducible.

In order to prove Theorem A, we need another property of the p-specialcharacters, of independent interest.

THEOREM B. Suppose that G is sol able of odd order. Let H be aŽ . Gsubgroup of G and let a g Irr H be a p-special character such that a is

< <irreducible. If J is a subgroup of G with H : H l J a p 9-number, thenŽ . J Ž .a g Irr J .H l J

† E-mail: gabriel@uv.es.* Research partially supported by DGICYT.

203

0021-8693r97 $25.00Copyright Q 1997 by Academic Press

All rights of reproduction in any form reserved.

GABRIEL NAVARRO204

It is not surprising that inducing properties about p-special characters ingroups of odd order become false when solvable groups in general are

Ž .considered as with Theorems A and B above . This was already wellw xknown by Gadjendragadkar with respect to Clifford induction 1 . As in

this case, it is conceivable that, by introducing linear factors, some versionsof Theorems A and B true may result for solvable groups. We have notattempted this, however.

We try to compensate for the odd hypotheses in Theorems A and B byintroducing a characteristic class of characters of p-separable groupswhich extend the Isaacs B -characters.p

Ž .THEOREM C. Suppose that G is a p-separable group and let x g B G .p

Ž . Ž . Ž .GIf H, a is a nucleus of x and b g Irr H is p 9-special then ab isŽ . Ž .G Ž .Girreducible. Furthermore, if g g Irr H is p 9-special, then ab s ag

if and only if b s g .

Since the Isaacs B -characters provide a canonical lift for the irreduciblep

Ž .Brauer characters when p s p9 , Theorems A and C above reveal anunexpected link between the p-special characters of certain subgroupswith the p-heights of the characters in the p-blocks of the group that, wethink, deserves further study.

2. PRELIMINARY RESULTS

We will begin by proving a stronger version of Theorem B which usesIsaacs p-theory heavily. For the reader’s convenience we recall a few factsabout p-partial characters.

If G is a p-separable group and G0 is the set of p-elements of G, ap-partial character w of G is a complex class function defined on G0 ofthe form x 0, where x is a character of G and x 0 is the restriction of x toG0. It is a nontrivial theorem to prove that the set of ‘‘irreducible’’

Žp-partial characters of G those which are not the sum of any two.p-partial characters is a basis of the complex space of class functions on

0 Ž .G . This basis is what we denote by I G . By the Fong Swan Theorem,p

Ž . Ž .notice that when p s p9, the set I G coincides with IBr G , the irre-p

ducible Brauer characters of G.Restriction, induction, and Clifford theory for I -characters are basicallyp

w xdeveloped in 4 .Ž .It is possible to find a canonical subset of Irr G which bijectively

Ž .lifts I G . Naturally associated to any irreducible character x of a p-p

w x Žseparable group G, M. Isaacs finds its ‘‘nucleus’’ in 3 . This is a unique up. Ž .to G-conjugacy pair W, g , where g is a p-factorable character of W

which induces x . The Isaacs B -characters, by definition, are those charac-p

p-SPECIAL CHARACTERS 205

ters of G such that its respective g is p-special. They turn out to be theŽ . Žcanonical lift for the set of irreducible p-partial characters I G Theo-p

Ž . w x.rem 9.3 of 3 . In this way, the p-special characters are naturallyextended to a larger class of canonical characters in p-separable groups.

Although the construction of the set B is somewhat complicated, thesep

characters are particularly easy to understand in groups of odd order.

Ž . Ž .2.1 THEOREM. Let G be a group of odd order and let x g Irr G . ThenŽ .x g B G if and only if x is induced from some p-special character.p

Proof. By definition, B -characters are always induced from p-specialp

Ž . w xcharacters. The converse follows from Corollary 3.4 of 6 , using the factthat p-special characters are B -characters.p

The oddness condition in Theorems A and B is not only due to thebetter behavior of the p-special and B -characters in groups of odd order.p

It comes from deeper reasons: the character theory associated to the oddfully ramified sections. We use the notion of controlled actions introduced

w xin 10 .

Ž .2.2 THEOREM. Suppose that KrN is an abelian odd normal section of Gand assume that the action of GrK on KrN is controlled. Let J be a

Ž .complement of KrN in G and assume that h g Irr N is G-in¨ariant andŽ .fully ramified with respect to KrN. If N : X : J and j g Irr X N h then

j K X is not irreducible.

ŽProof. By the theory of character triples see Theorem 11.28 andw x.Lemma 11.26 of 7 , we may assume that N is central and cyclic. Now, by

Ž . w xTheorem 3.1 of 2 , there exists a complement U of KrN in G such thatfor u, ¨ g V : U, u, ¨ are conjugate in VK if and only if they areconjugate in V. Since by hypothesis the action of GrK on KrN iscontrolled, we know that all complements of KrN in G are conjugate to J.Therefore, we may replace U by J and we have that for subgroupsN : X : J, two elements of X are conjugate in XK if and only if they are

Ž . w xconjugate in X. Now, by Lemma 4.3 of 3 , no character of X induces toXK irreducible.

Ž . Ž .3. PROOFS OF THEOREMS A AND B

First, we prove the stronger version of Theorem B.

Ž . Ž .3.1 THEOREM. Let G be a p-separable group with 2 f p . Let w g I Gp

Ž .and assume that a g I H has p-degree and induces w. If J is a subgroup ofp

< < Ž . J Ž .G with H : H l J a p 9-number, then a g I J .H l J p

GABRIEL NAVARRO206

< < < <Proof. We argue by induction, first on G , second on G : H and third< <on G : J .

Step 1. J is a maximal subgroup of G. If J - U - G, then H l J :< < < <H l U : H, and hence, H : H l U and H l U : H l J are p 9-< < Ž .U Ž .numbers. By induction on G : J , we have that a g I U . NowH lU p

U Ž .n s a has p-degree and induces n g I U . Also, the subgroup J ofH lU p

< Ž . < < <U is such that H l U : H l U l J s H l U : H l J is a p 9-numberŽ . J Ž . J Ž .and, hence, by induction on the order of G, a s n g I JH l J H l J p

as required.

Step 2. If N eG with N : H l J, then w is homogeneous. LetNŽ . Ž .u g I N be under a and suppose that T s I u - G. By the Cliffordp G

Ž Ž . w x.correspondence for I -characters see 3.2 of 4 , we may take j gp

Ž . H G TI T l H N u such that j s a . Hence, j s w and, in particular, j gp

Ž .I T .p

< <Since a has p-degree, it follows that H : T l H is a p-number and,Ž .Ž . < <therefore, H s H l T H l J . We conclude that T l H : T l J l H

is a p 9-number and thus the hypotheses of the theorem are satisfied insideT Ž .the group T with the character j g I T and the subgroup T l J of T.p

< <By induction on G , we have that

Tl Jj g I T l J .Ž . Ž .T l H l J p

Ž .T l JNow, notice that j lies over u , because j lies over u .T l H l J T l H l JThen, since T l J is the inertia group in J of the character u , by theClifford correspondence, we have that

JTlJ Jj s j g I J .Ž . Ž . Ž .Ž .T l H l J T l H l J p

Ž .H l J Ž H . Ž . J Ž .Since j s j s a , we see that a g I J ,T l H l J H l J H l J H l J p

as required.

< <Step 3. The index G : J is a p-number. If the index is not a p-number,since J is a maximal subgroup of a p-separable group G, then we have

< <that G : J is a p 9-number.First of all, observe that we may assume that w is not irreducible,J

Žbecause, otherwise, by taking any lift of a and applying Mackey ProblemŽ . w x. Ž . J5.7 of 7 , we would deduce that G s HJ. But in this case, a sH l J

Ž .w g I J , and this would prove the theorem.J p

ŽIn particular, we see that w does not have p-degree otherwise, since JŽ . w xcontains a Hall p-subgroup of G, by Corollary 2.6 of 4 , we will have that

Ž ..w g I J .J p

p-SPECIAL CHARACTERS 207

Ž . w xBy Proposition 3.4 of 4 , we may now find a normal subgroup M of Gsuch that all irreducible constituents of w have p-degree and are not G-M

Ž . w x < <invariant. By Lemma 3.4 of 9 , we have that MH : M is a p-number andtherefore, the character a M H has p-degree.

We claim that M : J. If this is not the case, by the maximality of J, we< <conclude that G s MJ. In particular, M : M l J is a p 9-number. There-

Ž . w xfore, by Corollary 2.6 of 4 , any irreducible constituent of w restrictsMŽ . w xirreducibly to M l J. Now, by Lemma 3.2 of 9 , we conclude that

Ž .w g I J and in this case we know that the theorem is proven.J p

Now, we notice that by the claim above and Step 2, we have thatŽ .MH ) H. Also, observe that J l MH s M J l H and then HM s

Ž . Ž . Ž .Ž .H J l MH and H l J l MH s H l M J l H s J l H. There-< < < <fore, HM : HM l J s H : H l J is a p 9-number and by induction on

< <G: H , we have that

JHMa g I J .Ž . Ž .Ž .HMlJ p

SinceJlM HHMa s a ,Ž . Ž .JlM H J l H

Ž . J Ž .then a g I J , and the proof of Step 3 is completed.H l J p

Ž . Ž . Ž .Step 4. core H s core J . Write N s core H . If N is not con-G G Gtained in J, then JN s G and thus G s HJ. But then, by Step 3, we have

< <that H : H l J is a p-number. This implies that G s J, and the theoremis certainly true in this case. So we may assume that N : J. Suppose that

Ž .N - S : core J , where SrN is a chief factor of G. We claim that SrNGis a p-group. Otherwise, we have that SrN is a p 9-group. Since N : H l J,

Ž .by Step 2, we may write w s eu , where u g I N is G-N p

invariant. Notice that u has p-degree, because it lies under a . Now, ifŽ . Ž .m g I S lies under w and necessarily over u , we have that m is ap N

Ž . w xmultiple of u . However, since SrN is a p 9-group by Lemma 3.1 of 4 , wehave that m extends u and, thus, it also has p-degree. Now, by LemmaŽ . w x < < < <3.4 of 9 , we have that HS : H s S : S l H is a p-number. SinceN : H l S, this is not possible. So we conclude that SrN is a p-number.

< < < < HSThen, 1 - S : S l H s HS : H is a p-number and hence a has p-Ž . < Ž . < < <degree. Now, J l SH s S J l H , and thus, HS : S H l J s H : H l J

is a p 9-number. By the argument used at the end of Step 3, the theorem< <follows by induction on G : H .

Step 5. Final contradiction. Let KrN be a chief factor of G. SinceŽ . < <G s JK by Step 4 and G : J is a p-number, we have that KrN is a

p-group. Hence, KrN is a group of odd order and, therefore, we have thatKrN is an abelian p-group for some prime p. Then, K l J s N. Since,

GABRIEL NAVARRO208

< <H : H l J is a p 9-number and K l HrN is a p-group, it follows thatK l H : H l J. Then K l H s K l H l J s H l N s N.

< <Now, notice that N - J. Otherwise, J eG and then H : H l J s< < Ž .HJ : J is a p-number by Step 3 and a p 9-number. Hence, H : J and inthis case the theorem is certainly true.

Now, we claim that the action of GrK on the abelian group KrN isŽ .controlled. To see this, first notice that C KrN s K, because N sG

Ž .core J . Now, we let MrK be a chief factor of G and we see thatGŽ .C MrK s 1. By the definition of controlled actions, it suffices toK r N

Ž < < < <.show that MrK , KrN s 1. This is of course true if MrK is a p 9-group.In the other case, where MrK is a p-group, by odd order, it follows thatMrK is an abelian q-group, for some prime q. However, it is not possible

Ž .that p s q, since in this case, MrN is a p-group and then Z MrN lŽ .KrN ) 1 implies that M : C KrN s K, a contradiction. This showsG

that the action of GrK on KrN is controlled.Ž . w x Ž .By Lemma 2.1 of 10 , we know that there is a unique t g Irr K N h

Ž . Žwhich is G-invariant, where h g B N lifts u notice that by the unique-p

.ness of the B -lifting, we have that h is G-invariant . Since KrN is anp

Ž . w xabelian chief factor of G, by the Going Down Theorem 6.18 of 7 , we2 < <have that t s h or that t s eh with e s K : N .N N

Ž . GLet j g Irr H be a p-special lifting of a and notice that j isirreducible, because j G lifts a G, which is irreducible. Also notice that jnecessarily lies over h, because normal irreducible constituents of p-

ŽŽ . w x.specials are p-specials 4.1 of 1 and the uniqueness of the B -lifting.pK H Ž K H . KHence, j is irreducible and j s uh contains t .K

Ž . w x Ž HK .If t s h, Corollary 4.2 of 3 implies j to be irreducible andN Hthis is impossible. So, we have that t is fully ramified with respect to KrN.

< Ž . < < < K HNow, since KH : K H l J s H : H l J is a p 9-number and a hasp-degree, it follows that

Ž .K HlJK Ha s a g I K H l J .Ž . Ž . Ž .Ž .Ž .K HlJ H l J p

Ž .K ŽH l J . Ž Ž ..Hence, j g Irr K H l J . Since N : H l J : J and j liesH l JŽ .over h, this contradicts Theorem 2.2 .

Now we can give a proof of Theorem B.

Ž .Proof of Theorem B. If a g Irr H is p-special and induces irreducibly,Ž . G Ž . Ž G.0we know by Theorem 2.1 that a g B G . Hence, w s a sp

Ž 0.G Ž . Ž . ŽŽ 0. . J ŽŽ . J .0 Ž .a g I G . By Theorem 3.1 , a s a g I J andp H l J H l J pJŽ .hence, a is necessarily irreducible.H l J

By using Theorem B, it is now possible to prove Theorem A.

< <Proof of Theorem A. We argue by induction, first on G and second on< <G : H .

p-SPECIAL CHARACTERS 209

Ž . Ž . Ž .Let L s core H , let n g Irr L be under a , and let u g Irr L beGŽ . w xunder b , so that n is p-special and u is p 9-special by 4.1 of 1 . Let

Ž . Ž .I s I n , let J s I u , and by the Clifford correspondence, let j gG GŽ . Ž . H HIrr I l H N n and t g Irr J l H N u be such that j s a and t s b.

We certainly have that I l H has p-index in H and that J l H hasp 9-index, because a has p-degree and b has p 9-degree. Hence, H sŽ .Ž .I l H J l H .

w xSince G has odd order, by Theorem B of 5 , we know that j is p-specialand that t is p 9-special. Also, since I l J l H has p 9-index in I l H and

Ž . w xp-index in H l J, by 6.1 of 1 , it follows that j is p-special andI l J l Hthat t is p 9-special. Now,I l J l H

HH IlH Hj t s j t s jb s ab .Ž . Ž . Ž .Ž .I l J l H I l J l H I l J l H I l H

G G Ž . I Ž .Observe that j s a g Irr G and hence we have that j g Irr I .Ž . Ž .Also, the subgroup I l J of I is such that I l H l H l J has p 9-Ž . I l J Ž .index in I l H. By Theorem B, we know that j g Irr I l J .I l J l H

If I l J - G, by using the inductive hypothesis, we have thatŽ . I l Jj t is irreducible. However, this character certainly liesI l J l H I l J l Hover nu and notice that I l J is the inertia group of this irreducible

ŽŽ .character by the uniqueness of the Gajendragadkar decomposition 7.1w x.of 1 . But then, by the Clifford correspondence, we have that

Ž .G Ž .Gj t is irreducible and therefore so is ab . So we seeI l J l H I l J l Hthat it is no loss if we assume that n and u are G-invariant.

Now, let KrL be a chief factor of G and write U s KH ) H andV s K l H.

Suppose first that KrL is a p 9-group. Since a is p-special and a U gŽ . Ž . U Ž . Ž .Irr U , by Theorem 2.1 , we know that a g B U . Let h g Irr Kp

under a U and notice that h lies over n . Also, since K eU, by CorollaryŽ . w x Ž .7.5 of 3 , we have that h g B K . Since KrL is a p 9-group and n isp

Ž . Ž . w xK-invariant, by Corollaries 6.3 and 6.5 of 3 , we have that h s n andLthat h and n uniquely determine one another. In particular, h is G-

Ž . w xinvariant. But in this case, by Corollary 4.2 of 3 , we know that restric-Ž . Ž . Ž U .tion defines a bijection from Irr U N h ª Irr H N h . Therefore, a isV H

irreducible and this is not possible since U / H.Ž .So we have then that KrL is a p-subgroup. Since u g Irr L is p 9-

Ž . w xspecial and K-invariant, by 4.3 of 1 , we know that there is a uniqueˆ ˆŽ .p 9-special character u g Irr K such that u s u . By uniqueness, we seeL

ˆthat u is G-invariant. Since V e H, for the same reason, we may writeˆb s ee , where e is the unique p 9-special character of V over u . Since uV V

ˆŽ . w xlies over u and is p 9-special by 6.1 of 1 , it follows that u s e lies underV

GABRIEL NAVARRO210

Ž . w xb. Again, by Lemma 4.2 of 3 , we know that restriction is a bijectionˆ ˆ ˆŽ . Ž . Ž .Irr U N u ª Irr H N u and since b g Irr H N u , we may find a uniqueV V

ˆ ˆ ˆŽ .b g Irr U N u such that b s b.Hˆ ˆNow, we claim that b is p 9-special. Since b has p 9-degree, by Lemma

ˆŽ . w x Ž .5.4 of 3 , it suffices to check that b g B U . For this, we use the magicp 9

ˆ ˆw xfield automorphism defined in Section 3 of 6 . Since we know that u , u ,VŽand b are p 9-specials and therefore fixed by the magic field automor-

ˆ ˆ. Ž . Ž .phism by the uniqueness in the restriction map Irr U N u ª Irr H N u ,Vˆit follows that b is also fixed by the field automorphism and then, by

Ž . w xLemma 3.1 of 6 , the claim follows.U Ž . < <Now, write a s a g Irr U . Since U : H is a p-number, it follows thatˆ

w x Ža is p-special by Theorem B of 5 or if we wish, by using the magic fieldˆ. Ž .Gautomorphism . Also, we have that a is irreducible. By induction onˆ

ˆ G< < Ž .G : H , we have that ab is irreducible. But now,ˆGG U G GG U Uˆ ˆ ˆab s ab s ab s a b s abŽ . Ž . ˆŽ . Ž .Ž . Ž .Hž /

is irreducible, as required.

Ž . Ž .3.2 EXAMPLES. Write H s GL 2, 3 s QS, where Qe H is thequaternion of order 8, S is the symmetric group on three letters, and

Ž . Ž . QQ l S s 1. Let Z s Z H , let 1 / l g Irr Z , and write l s 2u , whereˆŽ .u g Irr Q is H-invariant. If N s Z = S and l s l = 1 , it is well knownS

ˆthat l is irreducible.Hˆ� 4 ²Ž .:Write p s 2 and observe that l is p-special. Now, let C s 1, 2 g

Ž . Ž .Syl S and let J s QC, so that J g Syl H . If U s J l N s Z = C,2 2ˆ J< < < < Ž .then N : J l N s N : U s 3 is a p 9-number. We claim that l is notU

ˆ J QŽ .irreducible. If n s l is irreducible, then n s l s 2u . This is impos-U QŽ . w xsible, by Corollary 6.19 of 7 . This shows that Theorem B is not true for

Ž .groups of even order even solvable .Now, let HrQ s S act on V s A = A = A, where A is cyclic of order 3,

by permuting the factors and let G s VH be the semidirect product. LetŽ . Ž .j s 1 = 1 = m, where 1 / m g Irr A . Clearly, T s I j s VJ. SinceA A G

ˆ Ž .TrV is a 2-group, we may find j g Irr T to be 3-special, extending j .Now, let d be the 3-standard sign character defined in Section 2 ofŽG : V J .w x5 .

ˆ ˆBecause j is linear, t s j is, by definition, also 3-special. We claimV UŽ .V N Ž .that b s d t is irreducible and 3-special. By Theorem 2.5.d ofŽV N : V U .

w x5 , we have that V is in the kernel of d . Hence, d t is anŽV N : V U . ŽV N : V U .irreducible character in the inertia group of j in VN lying over j .

Ž .V NTherefore, by the Clifford correspondence, we have that d t isŽV N : V U .w xirreducible. But now, by Theorem B of 5 , it is also 3-special because

t s d d t is 3-special.ŽV N : V U . ŽV N : V U .

p-SPECIAL CHARACTERS 211

Now, by the previous part of the example, we already know that theˆ Ž .2-special character a s l g Irr VNrV induces irreducibly to G. We

Ž .Gclaim that ab is not irreducible. Otherwise,

GG V NV Nˆ ˆd t l s d tlŽ . ž /ž /ŽV N : V U . ŽV N : V U . V Už /is irreducible and then so is

V Jˆd tl .ž /ŽV N : V U . V U

Ž .Ž .However, since G s VJ VN and VJ l VN s VU, we have thatŽ . Žd s d this may be computed directly or by applyingŽV N : V U . ŽG : V J . V U

Ž . w x.Lemma 2.3 of 8 . Thus, we see that

V Jˆ ˆd j lŽ .ŽG : V J . V U

ˆ V JŽ .is irreducible. This forces l to be irreducible and this contradicts theV Ufirst example.

4. PROOF OF THEOREM C

w xIn this section we use definitions, notation, and several results of 3 .

Ž . Ž .4.1 THEOREM. Suppose that G is a p-separable group and let x g B G .p

Ž . Ž . Ž .GIf W, g is a nucleus of x and d g Irr W is p 9-special then gd isirreducible.

< <Proof. We argue by induction on G . If x is p-special, by the defini-Ž . Ž .tion of the B -characters, we have that W, g s G, x and then xd isp

Ž . w xirreducible by Proposition 7.1 of 1 . So we may assume that x is notp-special.

Ž .By the construction of the B -characters, we may find S, u a subnor-p

Ž . Ž . Ž .mal maximal p-factorable pair below G, x such that S, u F W, g withŽ . Ž . Ž IGŽS, u .. IGŽS, u .W : I S, u and such that W, g g nuc g . Hence, g gG

Ž Ž ..B I S, u , by definition. Since x is not p-special, we know thatp GŽ . Ž . w xI S, u - G, by Lemma 4.5 of 3 .G

Ž .Now, let b g Irr S be an irreducible constituent of d . Since SeeW,SŽ .we know that b is p 9-special and therefore, that ub g Irr S .

Ž .We claim that S, ub is a subnormal maximal p-factorable pair, be-Ž . Ž .cause if S, ub - R, ae , where ReeG, a is p-special and e is

Ž . Ž .p 9-special, then we will necessarily have that S, u - R, a , contradict-Ž .ing the maximality of S, u .

GABRIEL NAVARRO212

Ž . IGŽS, u .By the inductive hypothesis, we know that gd is irreducible.Ž . IGŽS, u .Notice that this character lies over ub , because gd lies over gd ,

which lies over ub. Now, by the uniqueness of the GadjendragadkarŽŽ . w x.decomposition 7.1 of 1 , we have that

I S, ub s I S, u l I S, b : I S, u .Ž . Ž . Ž . Ž .G G G G

Ž Ž . . Ž . IGŽS, u .Hence, we may find n g Irr I S, ub N ub under gd .GŽ .Now, since S, ub is a maximal subnormal p-factorable pair, we know

Ž . w xby Theorem 4.4 of 3 that induction defines a bijection

Irr I S, ub N ub ª Irr G N ub .Ž . Ž .Ž .G

Then n G is irreducible and hence, so is n IGŽS, u .. But then, n IGŽS, u . sI ŽS, u .GŽ .gd and the result follows.

Now we prove uniqueness in the induction process above.

Ž . Ž . Ž .4.2 THEOREM. Suppose that W , g is a nucleus for x g B G , fori i i p

Ž . Ž .G Ž .Gi s 1, 2, and let d g Irr W be p 9-special. If g d s g d , then therei i 1 1 2 2Ž . x Ž .exists x g G such that W , g d s W , g d .1 1 1 2 2 2

< <Proof. We argue by induction on G .Ž .As before, we choose S , u a maximal subnormal p-factorable pairi i

Ž . Ž . Ž . Ž . Ž Ž ..with S, u F G, x such that S , u F W , g F I S , u and withi i i i i i G i iŽ . Ž IGŽSi, u i..W , g g nuc g .i i i

Ž .G Ž .G Ž .Write x s g d s g d and let b g Irr S be an irreducible1 1 2 2 i iŽ . Ž .constituent of d . By the proof in Theorem 4.1 , we know that S , u biS i i ii

Ž .is a maximal subnormal p-factorable pair under G, x . Hence, byŽ . w xTheorem 3.2.b of 3 , there exists x g G such that

xS , u b s S , u b .Ž . Ž .1 1 1 2 2 2

Hence, S x s S , u x s u , and b x s b , by the uniqueness of the1 2 1 2 1 2Gadjendragadkar decomposition. Therefore, x is p-special iff S s G iff1 1S s G iff x is p-special. So, if this is the case, then2 2

x s x d s x d1 1 2 2

and the result again follows from the uniqueness of the Gajendragadkardecomposition.

Ž . Ž .So we may assume that I S ,u - G. Notice that I S , u sG 2 2 G 2 2Ž . xI S , u . Now, we have thatG 1 1

Ž .I S , ux x x G 2 2S , u F W , g F I S , u , gŽ . Ž .Ž . Ž .ž /2 2 1 1 G 2 2 1

p-SPECIAL CHARACTERS 213

andŽ .I S , ux x x G 2 2W , g g nuc g .Ž . Ž .ž /1 1 1

Ž x x. IGŽS2 , u 2 . Ž Ž . .Since g d lies over u b , we may find n gIrr I S , u N u b1 1 2 2 1 G 2 2 2 2Ž x x. IGŽS2 , u 2 . Ž Ž . .under g d and also, we may find n g Irr I S , u N u b1 1 2 G 2 2 2 2Ž . IGŽS2 , u 2 .under g d .2 2

Ž . w x GSince by Theorem 4.4 of 3 , n is irreducible, we have thati

GŽ . G GI S , uG x x GG 2 2n s g d s g d s g d s n .Ž . Ž .Ž .ž /1 1 1 1 1 2 2 2

Ž . w xBy the uniqueness in Theorem 4.4 of 3 , we deduce that n s n . But1 2Ž x x x. y Ž .now, by the inductive hypothesis, we have that W , g d s W , g d ,1 1 1 2 2 2

Ž ..for some y g N S , u and this completes the proof of the theorem.G 2 2

Now we can prove the uniqueness as stated in Theorem C of theIntroduction.

Ž .4.3 COROLLARY. Suppose that G is a p-separable group and let x gŽ . Ž . Ž .B G . Suppose that W, g is a nucleus of x and let m, d g Irr W bep

Ž .G Ž .Gp 9-specials. If gm s gd then m s d .

Ž .Proof. By Theorem 4.2 , we know that there is x g G such thatŽ . x Ž . Ž . x xW, gd s W, gm . Therefore, x g N W , g s g , and d s m. SinceG

NGŽW . Ž . w xg is irreducible, by Problem 6.1 of 7 , we have that x g W. Then,d s m, as required.

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7. I. M. Isaacs, ‘‘Character Theory of Finite Groups,’’ Dover, New York, 1994.8. I. M. Isaacs, Induction and restriction of p-partial characters and their lifts, to appear.9. I. M. Isaacs and G. Navarro, Weights and vertices for characters of p-separable groups,

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