aa r X i v : . [ m a t h . R T ] D ec WEIGHTS AND NILPOTENT SUBGROUPS
GABRIEL NAVARRO AND BENJAMIN SAMBALE
Abstract.
In a finite group G , we consider nilpotent weights, and prove a π -version of the Alperin Weight Conjecture for certain π -separable groups. Thiswidely generalizes an earlier result by I. M. Isaacs and the first author. Introduction
Let G be a finite group and let p be a prime. The celebrated Alperin WeightConjecture asserts that the number of conjugacy classes of G consisting of elementsof order not divisible by p is exactly the number of G -conjugacy classes of p -weights.Recall that a p -weight is a pair ( Q, γ ), where Q is a p -subgroup of G and γ ∈ Irr( N G ( Q ) /Q ) is an irreducible complex character with p -defect zero (that is, suchthat the p -part γ (1) p = | N G ( Q ) /Q | p ).In the main result of this paper, we replace p by a set of primes π as follows: Theorem A.
Let G be a π -separable group with a solvable Hall π -subgroup. Thenthe number of conjugacy classes of π ′ -elements of G is the number of G -conjugacyclasses of pairs ( Q, γ ) , where Q is a nilpotent π -subgroup of G and γ ∈ Irr( N G ( Q ) /Q ) has p -defect zero for every p ∈ π . Recall that a finite group is called π -separable if all its composition factors are π -groups or π ′ -groups. Let us restate Theorem A in the (presumably trivial) casewhere G itself is a (solvable) π -group. In this case, there is only one conjugacy classof π ′ -elements of G . On the other hand, if Q is a nilpotent subgroup of G , then γ ∈ Irr( N G ( Q ) /Q ) has p -defect zero for every p ∈ π if and only if N G ( Q ) = Q .Amazingly enough, there is only one conjugacy class of self-normalizing nilpotentsubgroups: the Carter subgroups of G (see p. 281 in [R]).Of course, if π = { p } , then Theorem A is the p -solvable case of the Alperin WeightConjecture (AWC). As a matter of fact, AWC was proven for π -separable groups witha nilpotent Hall π -subgroup by Isaacs and the first author [IN]. Now we realize that Mathematics Subject Classification.
Primary 20C15; Secondary 20C20.
Key words and phrases.
Alperin Weight Conjecture, Nilpotent subgroups, Carter subgroups.The research of the first author is supported by MTM2016-76196-P and Prometeo/GeneralitatValenciana. The second author thanks the German Research Foundation (projects SA 2864/1-1 andSA 2864/3-1). the nilpotency hypothesis can be dropped if one counts nilpotent weights instead.The solvability hypothesis is still needed, as shown by G = A and π = { , , } .There is a price to pay, however. The proof in [IN] relied on the so called Okuyama–Wajima argument, a definitely non-trivial but accessible tool on extensions of Glauber-man correspondents. In order to prove Theorem A, however, we shall need to appealto a deeper theorem of Dade and Puig (which uses Dade’s classification of the endo-permutation modules).As it is often the case in a “solvable” framework, the equality of cardinalities inTheorem A has a hidden structure which we are going to explain now. For sake ofconvenience we interchange from now on the roles of π and π ′ (of course, π -separableis equivalent to π ′ -separable). Recall that in a π -separable group G , the set I π ( G ) ofirreducible π -partial characters of G is the exact π -version (when π is the complementof a prime p ) of the irreducible Brauer characters IBr( G ) of a p -solvable group (seenext section for precise definitions). Each ϕ ∈ I π ( G ) has canonically associated a G -conjugacy class of π ′ -subgroups Q , which are called the vertices of ϕ . If I π ( G | Q )is the set of irreducible π -partial characters with vertex Q , unless π = p ′ , it is notin general true that | I π ( G | Q ) | = | I π ( N G ( Q ) | Q ) | . Instead we will prove the followingtheorem. Theorem B.
Suppose that G is π -separable with a solvable Hall π -complement. Let R be a nilpotent π ′ -subgroup of G and let Q be the set of π ′ -subgroups Q of G suchthat R is a Carter subgroup of Q . Then (cid:12)(cid:12)(cid:12) [ Q ∈Q I π ( G | Q ) (cid:12)(cid:12)(cid:12) = | I π ( N G ( R ) | R ) | . Since | I π ( N G ( R ) | R ) | is just the number of π ′ -weights with first component R (seeLemma 6.28 of [I2]), Theorem B implies Theorem A.As happens in the classical case where π = p ′ , and following the ideas of Dade,Kn¨orr and Robinson, one can define chains of π ′ -subgroups and relate them with π -defect of characters. This shall be explored elsewhere. Similarly, one can attachevery weight to a π ′ -block B of G by using Slattery’s theory [S]. In this setting weexpect that the number of π -partial characters belonging to B equals the number ofnilpotent weights attached to B .The groups described in Theorem A are sometimes called π -solvable. We did notfind a counterexample in the wider class of so-called π -selected groups. Here, π - selected means that the order of every composition factor is divisible by at mostone prime in π . P. Hall [H] has shown that these groups still have solvable Hall π -subgroups. Since every finite group is p -selected for every prime p , this version ofthe conjecture includes AWC in full generality.Unfortunately, Theorem A does not hold for arbitrary groups even if they possessnilpotent Hall π -subgroups. It is not so easy to find a counterexample, though. EIGHTS AND NILPOTENT SUBGROUPS 3
The fourth Janko group G = J has a cyclic Hall π -subgroup of order 35 (thatis, π = { , } ). The normalizers of the non-trivial π -weights are contained in amaximal subgroup M of type 2 . ( S × L (2)). However, l ( G ) − k ( G ) = 25 = 30 = l ( M ) − k ( M ), where l ( G ) denotes the number of π ′ -conjugacy classes and k ( G ) isthe number of π -defect zero characters of G . (The fact that J was a counterexamplefor the π -version of the McKay conjecture for groups with a nilpotent Hall π -subgroupwas noticed by Pham H. Tiep and the first author.)We take the opportunity to thank the developers of [GAP]. Without their tremen-dous work the present paper probably would not exist.The paper is organized as follows: In the next section we review π -partial characterswhich were introduced by Isaacs. In Section 3 we present two general lemmas oncharacters in π -separable groups. Afterwards we prove Theorem B. In the final sectionwe construct a natural bijection explaining Theorem B in the presence of a normalHall π -subgroup. 2. Review of π -theory Isaacs’ π -theory is the π -version in π -separable groups of the p -modular represen-tation theory for p -solvable groups. When π = p ′ , the complement of a prime, then I π ( G ) = IBr( G ) and we recover most of the well-known classical results. In whatfollows G is a finite π -separable group, where π is a set of primes. All the referencesfor π -theory can now be found together in Isaacs’ recent book [I2]. For the reader’sconvenience, we review some of the main features. If n is a natural number and p isa prime, recall that n p is the largest power of p dividing n . If π is a set of primes,then n π = Q p ∈ π n p . The number n is a π -number if n = n π .If G is a π -separable group, then G is the set of elements of G whose order isa π -number. A π -partial character of G is the restriction of a complex character of G to G . A π -partial character is irreducible if it is not the sum of two π -partialcharacters. We write I π ( G ) for the set of irreducible π -partial characters of G . Noticethat if µ ∈ I π ( G ) by definition there exists χ ∈ Irr( G ) such that χ = µ , where χ denotes the restriction of χ to the π -elements of G . Also, it is clear by the definition,that every π -partial character is a sum of irreducible π -partial characters. Noticethat if G is a π -group, then I π ( G ) = Irr( G ). Theorem 2.1 (Isaacs) . Let G be a finite π -separable group. Then I π ( G ) is a basisof the space of class functions defined on G . In particular, | I π ( G ) | is the number ofconjugacy classes of π -elements of G .Proof. This is Theorem 3.3 of [I2]. (cid:3)
GABRIEL NAVARRO AND BENJAMIN SAMBALE
We can induce and restrict π -partial characters in a natural way. If H is a subgroupof G and ϕ ∈ I π ( G ), then ϕ H = P µ ∈ I π ( H ) a µ µ for some uniquely defined nonnegativeintegers a µ . We write I π ( G | µ ) to denote the set of ϕ ∈ I π ( G ) such that a µ = 0.A non-trivial result is that Clifford’s theory holds for π -partial characters. If N ⊳ G ,it is then clear that G naturally acts on I π ( N ) by conjugation. Theorem 2.2 (Isaacs) . Suppose that G is π -separable and N ⊳ G .(a) If ϕ ∈ I π ( G ) , then ϕ N = e ( θ + · · · + θ t ) , where θ , . . . , θ t are all the G -conjugatesof some θ ∈ I π ( N ) .(b) If θ ∈ I π ( N ) and T = G θ is the stabilizer of θ in G , then induction defines abijection I π ( T | θ ) → I π ( G | θ ) .Proof. See Corollary 5.7 and Theorem 5.11 of [I2]. (cid:3)
In part (b) of Theorem 2.2, if µ G = ϕ , where µ ∈ I π ( T | θ ), then µ is called the Clifford correspondent of ϕ over θ , and sometimes it is written µ = ϕ θ .It is not a triviality to define vertices for π -partial characters (a concept that inclassical modular representation theory has little to do with character theory). Thiswas first accomplished in [IN] (generalizing a result of Huppert on Brauer charactersof p -solvable groups). Theorem 2.3.
Suppose that G is π -separable, and let ϕ ∈ I π ( G ) . Then there exist asubgroup U of G and α ∈ I π ( U ) of π -degree such that α G = ϕ . Furthermore, if Q isa Hall π -complement of U , then the G -conjugacy class of Q is uniquely determinedby ϕ .Proof. This is Theorem 5.17 of [I2]. (cid:3)
The uniquely defined G -class of π ′ -subgroups Q associated to ϕ by Theorem 2.3 iscalled the set of vertices of ϕ . If Q is a π ′ -subgroup of G , then we write I π ( G | Q ) todenote the set of ϕ ∈ I π ( G ) which have Q as a vertex. By definition, notice in thiscase that ϕ (1) π ′ = | G : Q | π ′ . Our last important ingredient is the Glauberman correspondence.
Theorem 2.4 (Glauberman) . Let S be a finite solvable group acting via automor-phisms on a finite group G such that ( | S | , | G | ) = 1 . Then there exists a canonicalbijection, called the S -Glauberman correspondence , Irr S ( G ) → Irr( C ) , χ χ ∗ , where Irr S ( G ) is the set of S -invariant irreducible characters of G and C = C G ( S ) .Here, χ ∗ is a constituent of the restriction χ C . Also, if T ⊳ S , then the T -Glaubermancorrespondence is an isomorphism of S -sets.Proof. See Theorem 13.1 of [I1]. (cid:3)
EIGHTS AND NILPOTENT SUBGROUPS 5 Preliminaries If G is a finite group, π is a set of primes, and χ ∈ Irr( G ), then we say that χ has π -defect zero if χ (1) π = | G | π . Lemma 3.1. If χ ∈ Irr( G ) has π -defect zero, then O π ( G ) = 1 .Proof. Let
N ⊳ G , and let θ ∈ Irr( N ) be under χ . Then we have that θ (1) divides | N | and χ (1) /θ (1) divides | G : N | by Corollary 11.29 of [I1]. Thus χ (1) π = | G | π if andonly if θ (1) π = | N | π and ( χ (1) /θ (1)) π = | G : N | π . The result is now clear applyingthis to N = O π ( G ). (cid:3) We shall use the following notation. Suppose G is π -separable, N ⊳ G , τ ∈ I π ( N )and Q is a π ′ -subgroup of G . Then I π ( G | Q, τ ) = I π ( G | Q ) ∩ I π ( G | τ ) . Lemma 3.2.
Suppose G is π -separable and that N ⊳ G . Let Q be a π ′ -subgroup of G .(a) Suppose that µ ∈ I π ( G | Q ) . Then there is a unique N G ( Q ) -orbit of τ ∈ I π ( N ) such that µ τ ∈ I π ( G τ | Q ) , where µ τ is the Clifford correspondent of µ over τ .Every such τ is Q -invariant.(b) Suppose that τ ∈ I π ( N ) is Q -invariant. Let U be a complete set of representativesof the G τ -orbits on the set { Q g | g ∈ G, Q g ⊆ G τ } . Then | I π ( G | Q, τ ) | = X U ∈U | I π ( G τ | U, τ ) | . Thus, if G τ N G ( Q ) = G , then | I π ( G | Q, τ ) | = | I π ( G τ | Q, τ ) | . Proof. (a) Let ν ∈ I π ( N ) be under µ , and let µ ν ∈ I π ( G ν | ν ) be the Clifford correspon-dent of µ over ν . If R is a vertex of µ ν , then R is a vertex of µ , by Theorem 2.3.Therefore R = Q g for some g ∈ G . If τ = ν g − , then we have that µ τ has vertex Q . Suppose now that ρ ∈ I π ( N ) is under µ such that µ ρ has vertex Q . ByTheorem 2.2(a), there exists g ∈ G such that τ g = ρ . Thus Q g is a vertex of µ ρ .Then there is x ∈ G ρ such that Q gx = Q . Since τ gx = ρ , the proof of part (a) iscomplete.(b) We have that induction defines a bijection I π ( G τ | τ ) → I π ( G | τ ). Notice that [ U ∈U I π ( G τ | U )is a disjoint union. It suffices to observe, again, that if ξ ∈ I π ( G τ | τ ) has vertex U , then ξ G has vertex U . (cid:3) GABRIEL NAVARRO AND BENJAMIN SAMBALE Proofs
The deep part in our proofs comes from the following result.
Theorem 4.1.
Suppose that L is a normal π -subgroup of G , Q is a solvable π ′ -subgroup of G such that LQ ⊳ G . Suppose that M ⊆ Z ( G ) is contained in L and that ϕ ∈ Irr( M ) . Then | I π ( G | Q, ϕ ) | = | I π ( N G ( Q ) | Q, ϕ ) | .Proof. Let A be a complete set of representatives of N G ( Q )-orbits on Irr Q ( L | ϕ ), the Q -invariant members of Irr( L | ϕ ). Using Lemma 3.2, we have that I π ( G | Q, ϕ ) = [ τ ∈A I π ( G | Q, τ )is a disjoint union. Let A ∗ be the set of the Q -Glauberman correspondents of theelements of A . Notice that A ∗ is a complete set of representatives of N G ( Q )-orbitson Irr( C L ( Q ) | ϕ ). Moreover, N G τ ( Q ) = N G τ ∗ ( Q ). Then, as before, I π ( N G ( Q ) | Q, ϕ ) = [ τ ∈A I π ( N G ( Q ) | Q, τ ∗ )is a disjoint union. Thus | I π ( N G ( Q ) | Q, ϕ ) | = X τ ∈A | I π ( N G τ ( Q ) | Q, τ ∗ ) | . Thus we need to prove that | I π ( G τ | Q, τ ) | = | I π ( N G τ ( Q ) | Q, τ ∗ ) | . We may assume that τ is G -invariant.Now, since LQ ⊳ G and τ is G -invariant, by Lemma 6.30 of [I2], we have that Q is contained as a normal subgroup in some vertex of θ , whenever θ ∈ I π ( G ) lies over τ . Therefore θ ∈ I π ( G | τ ) has vertex Q if and only if θ (1) π ′ = | G : Q | π ′ . Similarly, θ ∈ I π ( N G ( Q ) | τ ∗ ) has vertex Q if any only if θ (1) π ′ = | N G ( Q ) : Q | π ′ = | G : Q | π ′ since G = L N G ( Q ) by the Frattini argument and the Schur–Zassenhaus theorem.Now we use the Dade–Puig theory on the character theory above Glaubermancorrespondents, which is thoroughly explained in [T]. By Theorem 6.5 of [T], in thelanguage of Chapter 11 of [I1] (see Definition 11.23 of [I1]), we have that the char-acter triples ( G, L, τ ) and ( N G ( Q ) , C L ( Q ) , τ ∗ ) are isomorphic. Write ∗ : Irr( G | τ ) → Irr( N G ( Q ) | τ ∗ ) for the associated bijection of characters. By Lemma 6.21 of [I2], thereexists a unique bijection ∗ : I π ( G | τ ) → I π ( N G ( Q ) | τ ∗ )such that if χ = φ ∈ I π ( G | τ ) and χ ∈ Irr( G ) (which necessarily lies over τ ), then( χ ∗ ) = φ ∗ . Since χ (1) /τ (1) = χ ∗ (1) /τ ∗ (1) (by Lemma 11.24 of [I1]), it follows that EIGHTS AND NILPOTENT SUBGROUPS 7 χ (1) π ′ = χ ∗ (1) π ′ . We deduce that | I π ( G | Q, τ ) | = | I π ( N G ( Q ) | Q, τ ∗ ) | , as desired. (cid:3) In order to prove Theorem B, we argue by induction on the index of a normal π -subgroup M of G . Theorem B follows from the special case M = 1. Theorem 4.2.
Suppose that G is π -separable with a solvable Hall π -complement.Let R be a nilpotent π ′ -subgroup of G . Let M ⊳ G be a normal π -subgroup, and let ϕ ∈ Irr( M ) be G -invariant. Let Q be the set of π ′ -subgroups Q of G such that R isa Carter subgroup of Q . Then (cid:12)(cid:12)(cid:12) [ Q ∈Q I π ( G | Q, ϕ ) (cid:12)(cid:12)(cid:12) = | I π ( M N G ( R ) | R, ϕ ) | . Proof.
We argue by induction on | G : M | .By Lemma 3.11 of [I2], let ( G ∗ , M ∗ , ϕ ∗ ) be a character triple isomorphic to ( G, M, ϕ ),where M ∗ is a central π -subgroup of G ∗ . If Q is a π ′ -subgroup of G , notice that wecan write ( QM ) ∗ = M ∗ × Q ∗ , for a unique π ′ -subgroup Q ∗ of G ∗ . If R is contained ina π ′ -subgroup Q , then R is a Carter subgroup of Q if and only if RM/M is a Cartersubgroup of
QM/M , using that Q is naturally isomorphic to QM/M . This happensif and only if (
RM/M ) ∗ is a Carter subgroup of ( QM/M ) ∗ , which again happensif and only if R ∗ is a Carter subgroup of Q ∗ . Notice further that if R is a Cartersubgroup of Q , then R is a Carter subgroup of every Hall π -complement Q of QM that happens to contain R (again using the isomorphism between QM/M and Q ).We easily check now that the set of π ′ -subgroups of G ∗ that contain R ∗ as a Cartersubgroup is exactly Q ∗ = { Q ∗ | Q ∈ Q} .By the Frattini argument and the Schur–Zassenhaus theorem, notice that N G ( M R ) = M N G ( R ). By Lemma 6.21 and the proof of Lemma 6.32 of [I2], there is a bijection ∗ : I π ( G | ϕ ) → I π ( G ∗ | ϕ ∗ ) such that η has vertex Q if and only if η ∗ has vertex Q ∗ .From all these arguments, it easily follows that we may assume that M is central. Inparticular, M ≤ N G ( R ).Let K = O π ′ ( G ). Suppose that there exists some µ ∈ I π ( G | Q, ϕ ) for some Q ∈ Q .By Lemma 6.30 of [I2] (in the notation of that lemma, K is 1 and Q is K ), we havethat K is contained in Q . Hence, it is no loss if we only consider Q ∈ Q such that K ⊆ Q .Suppose that N K ( R ) is not contained in R . Then there cannot be weights ( R, γ ),where γ ∈ Irr( N G ( R ) /R ) has π -defect zero by Lemma 3.1. So the right hand side iszero. Suppose that there exists some µ ∈ I π ( G | Q, ϕ ) for some Q ∈ Q (with K ⊆ Q ).Since R is a Carter subgroup of Q , then R is a Carter subgroup of KR , and therefore N K ( R ) is contained in R . Therefore may assume that N K ( R ) is contained in R . Weclaim that R is a Carter subgroup of Q if and only if RK/K is a Carter subgroup of
GABRIEL NAVARRO AND BENJAMIN SAMBALE
Q/K . One implication is known (see 9.5.3 in [R]). Suppose that
RK/K is a Cartersubgroup of
Q/K . Since N Q ( R ) normalizes RK , it is contained in RK . Hence N Q ( R ) = N KR ( R ) = R , and R is a Carter subgroup of Q . In this situation theFrattini argument yields N G ( R ) K = N G ( RK ).Next, we will replace G by G/K . By Lemma 6.31 of [I2] (the roles of K and M are interchanged in that lemma), | I π ( G | Q, ϕ ) | = | I π ( G/K | Q/K, ˆ ϕ ) , where ˆ ϕ ∈ Irr(
M K/K ) corresponds to ϕ via the natural isomorphism. Similarly, | I π ( N G ( R ) | R, ϕ ) | = | I π ( N G ( R ) K/K | RK/K, ˆ ϕ ) | = | I π ( N G ( RK ) /K | RK/K, ˆ ϕ ) | . Hence, for the remainder of the proof we may assume that O π ′ ( G ) = K = 1.Suppose now that L = O π ( G ). Let A be a complete set of N G ( R )-representativesof the R -invariant characters in Irr( L | ϕ ). If L = M , then L = G by the Hall-HigmanLemma 1.2.3, and G is a π -group. In this case, R = 1 = Q , and there is nothing toprove. Thus, we may assume that | G : L | < | G : M | .For each τ ∈ A , let Q τ be the set of π ′ -subgroups Q of G τ such that R is a Cartersubgroup of Q . By induction, (cid:12)(cid:12)(cid:12) [ Q ∈Q τ I π ( G τ | Q, τ ) (cid:12)(cid:12)(cid:12) = | I π ( L N G τ ( R ) | R, τ ) | . Since L is a π -group and R is a π ′ -subgroup, we have that L N G ( R ) = N G ( LR ). Also, N G ( LR ) τ = L N G ( R ) τ . By Lemma 3.2, we have that | I π ( L N G τ ( R ) | R, τ ) | = | I π ( L N G ( R ) | R, τ ) | . Also, | I π ( L N G ( R ) | R, ϕ ) | = X τ ∈A | I π ( L N G ( R ) | R, τ ) | , by using the first paragraph of the proof of Theorem 4.1. By Theorem 4.1, | I π ( L N G ( R ) | R, ϕ ) | = | I π ( N G ( R ) | R, ϕ ) | . Therefore, X τ ∈A (cid:12)(cid:12)(cid:12) [ Q ∈Q τ I π ( G τ | Q, τ ) (cid:12)(cid:12)(cid:12) = | I π ( N G ( R ) | R, ϕ ) | . We are left to show that (cid:12)(cid:12)(cid:12) [ Q ∈Q I π ( G | Q, ϕ ) (cid:12)(cid:12)(cid:12) = X τ ∈A (cid:12)(cid:12)(cid:12) [ Q ∈Q τ I π ( G τ | Q, τ ) (cid:12)(cid:12)(cid:12) . Let R be a complete set of representatives of N G ( R )-orbits in Q , and notice that [ Q ∈Q I π ( G | Q, ϕ ) = [ Q ∈R I π ( G | Q, ϕ ) EIGHTS AND NILPOTENT SUBGROUPS 9 is a disjoint union. Indeed, if µ ∈ I π ( G | Q , ϕ ) ∩ I π ( G | Q , ϕ ) for Q i ∈ Q , then wehave that Q = Q g for some g ∈ G by the uniqueness of vertices. Hence R g and R are Carter subgroups of Q , and therefore R gx = R for some x ∈ Q . It follows that Q = Q gx are N G ( R )-conjugate.Now fix Q ∈ R . For each µ ∈ I π ( G | Q, ϕ ), we claim that there is a unique τ ∈ A such that µ τ ∈ I π ( G τ | Q x , τ ), for some x ∈ N G ( R ). We know that there is ν ∈ Irr( L | ϕ )such that µ ν ∈ I π ( G ν | Q, ν ) by Lemma 3.2(a). Now, ν x = τ for some x ∈ N G ( R )and τ ∈ A , and it follows that µ τ ∈ I π ( G τ | Q x , τ ). Suppose that µ ǫ ∈ I π ( G ǫ | Q y , ǫ ),for some y ∈ N G ( R ) and ǫ ∈ A . Now, ǫ = τ g for some g ∈ G , by Clifford’stheorem. Thus Q xgt = Q y for some t ∈ G ǫ , by the uniqueness of vertices. Thus xgty − ∈ N G ( Q ). Since R is a Carter subgroup of Q , by the Frattini argument wehave that xgty − = qv , where q ∈ Q and v ∈ G normalizes Q and R . Since Q x fixes τ , then Q fixes τ x − . Now ǫ y − = ( τ gt ) y − = τ x − xgty − = τ x − qv = τ x − v . So ǫ and τ are N G ( R )-conjugate, and thus they are equal.Now we define a map f : [ Q ∈R I π ( G | Q, ϕ ) → [ τ ∈A [ Q ∈Q τ I π ( G τ | Q, τ ) ! × { τ } ! given by f ( µ ) = ( µ τ , τ ), where τ ∈ A is the unique element in A such that µ τ ∈ I π ( G τ | Q x , τ ), for some x ∈ N G ( R ). Since µ Gτ = µ , we have that f is injective. If wehave that γ ∈ S Q ∈Q τ I π ( G τ | Q, τ ) then γ G ∈ S Q ∈Q I π ( G | Q, ϕ ), so f is surjective. (cid:3) Some of the difficulties in Theorem 4.2 are caused by the fact that Clifford corre-spondence does not necessarily respect vertices, even in quite restricted situations.Suppose that N is a normal p ′ -subgroup of G , τ ∈ Irr( N ), Q is a p -subgroup of G and τ is Q -invariant. Then it is not necessarily true that induction defines a bijectionIBr( G τ | Q, τ ) → IBr( G | Q, τ ). For instance, take p = 2 and G = SmallGroup (216 , N of order 3. The Fitting subgroup F of G is F = N × M , where M is a normal subgroup of type C × C , and G/F = D . Let1 = τ ∈ Irr( N ). Then G τ ⊳ G has index 2, and G τ /N = S × S . Now τ has a uniqueextension ˆ τ ∈ IBr( G τ ). The group G τ has three conjugacy classes of subgroups Q of order 2. Take Q that corresponds to C × Q that corresponds to 1 × C .Then Q and Q are not G τ -conjugate but G -conjugate. So | IBr( G τ | Q , τ ) | = 1 and | IBr( G | Q , τ ) | = 2. 5. A canonical bijection If G has a normal Hall π -subgroup, then we have a canonical bijection in Theo-rem 4.2. This seems worth to be explored. Lemma 5.1.
Suppose that G = N H where N is a normal π -subgroup and H is a π ′ -subgroup. Then N G ( Q ) = C N ( Q ) N H ( Q ) for every Q ≤ H .Proof. First note that Q = Q ( N ∩ H ) = QN ∩ H ⊳ N N G ( Q ) ∩ H ≤ N H ( Q ) . Let xh ∈ N G ( Q ) where x ∈ N and h ∈ H . Then h = x − ( xh ) ∈ N N G ( Q ) ∩ H ≤ N H ( Q ). This shows N G ( Q ) = N N ( Q ) N H ( Q ) = C N ( Q ) N H ( Q ). (cid:3) Lemma 5.2.
Suppose that G = N H where N is a normal π -subgroup and H is asolvable π ′ -subgroup. Let R ≤ H , and let τ ∈ Irr( C N ( R )) be such that N G ( R ) τ = C N ( R ) × R . Let γ ∈ Irr R ( N ) be the Glauberman correspondent of τ . Then R = N H γ ( R ) .Proof. Suppose that
R < S ≤ H γ , where R ⊳ S . Then S acts on the R -Glaubermancorrespondence. Since S fixes γ , therefore it fixes γ ∗ = τ . But this gives the contra-diction S ⊆ N G ( R ) τ = C N ( R ) × R . (cid:3) Theorem 5.3.
Suppose that G = N H where N is a normal π -subgroup and H isa solvable π ′ -subgroup. Let R be a nilpotent subgroup of H . Let Q be the set ofsubgroups Q ⊆ H such that R is a Carter subgroup of Q . Then there is a naturalbijection [ Q ∈Q I π ( G | Q ) → I π ( N G ( R ) | R ) . Proof.
Let Q ∈ Q . By the Frattini argument, notice that N G ( Q ) = Q ( N G ( Q ) ∩ N G ( R )), and that Q ∩ ( N G ( Q ) ∩ N G ( R )) = R .Let φ ∈ I π ( G | Q ). By Lemma 3.2, there exists a Q -invariant θ ∈ Irr( N ) under φ . Then T = G θ = QN using Corollary 8.16 in [I1] for instance. If θ is anothersuch choice, then θ = θ g for some g ∈ N G ( Q ). Thus, we may assume that g ∈ N G ( Q ) ∩ N G ( R ). Let θ ∗ ∈ C N ( R ) be the R -Glauberman correspondent of θ . Now,by Lemma 5.1 applied in T , we have that N T ( R ) = C N ( R ) N Q ( R ) = C N ( R ) × R .We claim that N T ( R ) is the stabilizer of θ ∗ in N G ( R ). If x ∈ N G ( R ) fixes θ ∗ , then x fixes θ , and thus x ∈ N T ( R ), as claimed. Now φ ∗ := ( θ ∗ × R ) N G ( R ) is irreducible,and belongs to I π ( N G ( R ) | R ). Since θ is N G ( Q ) ∩ N G ( R )-conjugate to θ , φ ∗ isindependent of the choice of θ .Suppose that φ ∗ = µ ∗ , where φ ∈ I π ( G | Q ) and µ ∈ I π ( G | Q ), where R is a Cartersubgroup of Q i and Q i ⊆ H . Suppose that we picked θ for φ and ǫ for µ , so that φ ∗ = ( θ ∗ × R ) N G ( R ) and µ ∗ = ( ǫ ∗ × R ) N G ( R ) . Then N G ( R ) = C N ( R ) N H ( R ), and θ ∗ and ǫ ∗ are N H ( R )-conjugate, say ( θ ∗ ) x = ǫ ∗ . Then θ x = ǫ . By replacing ( Q , θ ) by EIGHTS AND NILPOTENT SUBGROUPS 11 ( Q x , θ x ), we may assume that θ = ǫ . But then Q = H θ = H ǫ = Q . Since π -partialcharacter are determined on the π -elements, we must have φ = µ now.Suppose conversely that τ ∈ I π ( N G ( R ) | R ). Then τ is induced from C N ( R ) × R .Let µ ∈ Irr( C N ( R )) such that µ × R induces τ . Then the stabilizer of µ in N G ( R )is C N ( R ) × R . If ρ ∈ Irr R ( N ) is the R -Glauberman correspondent of µ , then byLemma 5.2 we know that R is a Carter subgroup of Q = H ρ , where Q is the stabilizerin H of ρ . Thus with the notation of the first part of the proof we obtain τ = µ ∗ where µ is induced from G ρ = QN . (cid:3) References [GAP] The GAP group, ‘
GAP - groups, algorithms, and programming ’, Version 4.10.0, 2018, .[I1] I. M. Isaacs, ‘
Character Theory of Finite Groups ’, AMS Chelsea, Providence, RI, 2006.[I2] I. M. Isaacs, ‘
Characters of Solvable Groups ’, Graduate studies in Mathematics ,AMS, Providence, RI, 2018.[IN] I. M. Isaacs, G. Navarro, Weights and vertices for characters of π -separable groups, J.Algebra (1995), 339–366.[R] D. J. S. Robinson, ‘ A course in the theory of groups ’, 2nd edition, Springer-Verlag, NewYork, 1996.[H] P. Hall, Theorems like Sylow’s, Proc. London Math. Soc. (1956), 286–304.[S] M. C. Slattery, Pi-blocks of pi-separable groups. I, J. Algebra (1986), 60–77.[T] A. Turull, Above the Glauberman correspondence, Adv. in Math. (2008), 2170–2205. Department of Mathematics, Universitat de Val`encia, 46100 Burjassot, Val`encia,Spain
E-mail address : [email protected] Institut f¨ur Mathematik, Friedrich-Schiller-Universit¨at Jena, 07737 Jena, Ger-many
E-mail address ::