Brauer's Height Zero Conjecture for Principal Blocks
aa r X i v : . [ m a t h . R T ] F e b BRAUER’S HEIGHT ZERO CONJECTURE FOR PRINCIPAL BLOCKS
GUNTER MALLE AND GABRIEL NAVARRO
Abstract.
We prove the other half of Brauer’s Height Zero Conjecture in the case ofprincipal blocks. Introduction
Richard Brauer’s famous Height Zero Conjecture [2] from 1955 proposes that if p is aprime, G is a finite group, and B is a Brauer p -block of G with defect group D , then D is abelian if and only if all the irreducible complex characters of G in B have height zero.For abelian D , the proof was finally completed in [11]. The other direction was provento hold assuming that all simple groups satisfy the inductive Alperin–McKay condition (see Navarro–Sp¨ath [19]). Prior to this, the conjecture was shown for p = 2 and blocks ofmaximal defect by Navarro–Tiep [21].The inductive Alperin–McKay condition (Sp¨ath [23]) is a natural but difficult-to-checkstatement on blocks of simple groups which involves action of automorphisms and co-homology. In this paper, we take a step back and conduct a direct reduction to almostsimple groups of Brauer’s conjecture for principal blocks. Then we handle these groupsvia the classification of finite simple groups, hence proving the conjecture in this case.We denote by B ( G ) the principal p -block of G , and by Irr( B ( G )) the set of complexirreducible characters of G in B ( G ). Theorem A.
Let p be a prime and let G be a finite group. Then the Sylow p -subgroupsof G are abelian if and only if p does not divide χ (1) for all χ ∈ Irr( B ( G )) . In Problem 12 of his famous list [3] from 1963 Brauer asked if the character table ofa finite group G detects if G has abelian Sylow p -subgroups. Although this has beenpreviously solved in [13] (theoretically) and [20] (with an algorithm), Theorem A givesthe solution to Problem 12 that surely Brauer had in mind.A crucial ingredient in our proof is the result by Kessar–Malle [12] that Brauer’s HeightZero Conjecture holds for all quasi-simple groups.At several points of the proof we do use special properties of principal blocks, such asthe Alperin–Dade theory of isomorphic blocks, which do not hold true for general blocks. Date : February 17, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Brauer’s Height Zero Conjecture, principal blocks.The first author gratefully acknowledges financial support by SFB TRR 195. The research of thesecond author is supported by Ministerio de Ciencia e Innovaci´on PID2019-103854GB-I00 and FEDERfunds. We thank Yanjun Liu, Lizhong Wang, Wolfgang Willems and Jiping Zhang for e-mails promptingour interest in the subject of this paper, and Yanjun Liu for pertinent questions on an earlier version. Reduction to Almost Simple Groups
Theorem 2.1.
Brauer’s Height Zero Conjecture is true for principal p -blocks if it is truefor the principal p -block of every almost simple group S such that S/ soc( S ) is a p -group.Proof. Let G be a finite group, and let P ∈ Syl p ( G ). We assume that p does not dividethe degrees of the complex irreducible characters in B ( G ). We argue by induction on | G | that P is abelian.Let 1 < N < G be a normal subgroup of G . Since Irr( B ( G/N )) ⊆ Irr( B ( G )), we havethat G/N has abelian Sylow p -subgroups, by induction. If θ ∈ Irr( B ( N )), then thereexists χ ∈ Irr( B ( G )) such that θ is an irreducible constituent of the restriction χ N (by[18, Thm. 9.4]). Since θ (1) divides χ (1), we have that N has abelian Sylow p -subgroups,by induction. In particular, we have that O p ′ ( G ) = G . If G has two different minimalnormal subgroups N and N , then G is isomorphic to a subgroup of G/N × G/N thathas abelian Sylow p -subgroups, and therefore G has abelian Sylow p -subgroups. So wemay assume that G has a unique minimal normal subgroup N .Since Irr( B ( G )) = Irr( B ( G/ O p ′ ( G )) (by [18, Thm. 6.10]), we may assume by induc-tion that O p ′ ( G ) = 1.Recall that if a finite group H has abelian Sylow p -subgroups and O p ′ ( H ) = 1, then O p ′ ( H ) = O pp ′ ( H ) × O p ( H ), where O pp ′ ( H ) = O p ( O p ′ ( H )) is either trivial or the di-rect product of non-abelian simple groups of order divisible by p , with abelian Sylow p -subgroups, and O p ( H ) is abelian. (See for instance, [22, Thm. 2.1].) Hence, if M is anyproper normal subgroup of G , then O p ′ ( M ) is a direct product of O p ( M ) and O p ′ p ( M ),and this latter is either trivial or a direct product of simple groups of order divisible by p . Also, again, if L/N = O p ′ ( G/N ), we have that
G/L = X/L × Y /L , where
X/L is adirect product of simple groups of order divisible by p (or trivial), and Y /L is a p -group.Suppose first that N is a central p -group. Let τ ∈ Irr( P ). Write τ N = eλ , where λ ∈ Irr( N ). By [18, Thm. 9.4], let γ ∈ Irr( B ( N G ( P ))) over τ . By [18, Cor. 6.4] andBrauer’s Third Main theorem, there is some ˆ τ ∈ Irr( B ( G )) over γ . Thus ˆ τ has p ′ -degree.Hence, ˆ τ P contains some p ′ -degree character γ , and γ N = λ ∈ Irr( N ) by [9, Cor. 11.29].Since P/N is abelian by induction, by Gallagher’s Corollary 6.17 of [9] we have τ = γρ for some linear ρ ∈ Irr(
P/N ). Hence, all irreducible characters of P are linear, and weare done in this case.Suppose next that N is a p -group. Let C = C G ( N ). Then all characters in Irr( G/C )are in the principal p -block of G (by [21, Lemma 3.1] and Brauer’s third main theorem).By the Itˆo–Michler theorem [17] and our hypothesis, we have that G/C has a normalSylow p -subgroup. Since O p ′ ( G ) = G , we have that G/C is a p -group. By the previousparagraph, we may assume that C < G . Hence, O p ′ ( C ) is a direct product of O p ′ p ( C )and O p ( C ). These are normal subgroups of G , since they are characteristic in C . Since G has a unique minimal normal subgroup and O p ( C ) >
1, we conclude that O p ′ ( C ) is a p -group. In this case, G is p -solvable and Irr( B ( G )) = Irr( G ) (by [18, Thm. 10.20], forinstance). Again by the Itˆo–Michler theorem, we conclude that G has a normal Sylow p -subgroup. Hence G = P as O p ′ ( G ) = G , and again we are done.Thus we may assume that O p ( G ) = 1. Hence, N = F ∗ ( G ) is a direct product of non-abelian simple groups of order divisible by p . Also, if M is any proper normal subgroup of G , then O p ′ ( M ) is a direct product of non-abelian simple groups of order divisible by p . RAUER’S HEIGHT ZERO CONJECTURE FOR PRINCIPAL BLOCKS 3
Let Q = P ∩ N ∈ Syl p ( N ). We have that the irreducible characters of Irr( G/N C G ( Q ))are in the principal block (again by [21, Lemma 3.1] and Brauer’s third main theorem), so G/N C G ( Q ) is a p -group by the Itˆo–Michler theorem, and using that O p ′ ( G ) = G . Then L ⊆ N C G ( Q ). Hence, L = N C L ( Q ).Since X is not a p ′ -group (because O p ′ ( G ) = 1), we have that N ⊆ O p ′ ( X ). Assume that X < G . Since O p ′ ( X ) is a direct product of non-abelian simple groups and N E O p ′ ( X ), wehave O p ′ ( X ) = N × U , by elementary group theory. But then U = 1, because N = F ∗ ( G )and C G ( N ) ⊆ N . Thus O p ′ ( X ) = N . Since X/L is a direct product of non-abeliansimple groups of order divisible by p (or trivial) and N ⊆ L ⊆ X , we deduce that X = L and therefore Y = G . Thus G/N has a normal p -complement L/N . We claim that
P N satisfies the hypotheses. Let τ ∈ Irr( B ( P N )). Let γ ∈ Irr( N ) be under τ . Then γ liesin the principal block of N . Since τ lies under some character in the principal block of G , we have that γ has p ′ -degree using the hypothesis. By the Alperin–Dade theory onisomorphic blocks ([1, 6]), γ has a canonical extension ˆ γ ∈ Irr( L ) in the principal block of L . Now, ˆ γ lies under some χ ∈ Irr( B ( G )). By hypothesis, χ has p ′ -degree, and therefore χ L = ˆ γ (by [9, Cor. 11.29]), and ˆ γ is G -invariant. In particular, γ is G -invariant too. Bythe Isaacs character correspondence ([8, Cor. 4.2]) there is a bijectionIrr( G | ˆ γ ) → Irr(
P N | γ ) , given by restriction. Since τ lies over γ , let ˆ τ ∈ Irr( G ) over ˆ γ that restricts to τ . Since G/L is a p -group and ˆ τ lies over ˆ γ , we have that ˆ τ is in the principal p -block of G ([18,Cor. 9.6]), thus τ has p ′ -degree. So, by induction, we may assume that G = N P . Let S benormal simple in N and write N = S x × · · · × S x t , where P = N P ( S ) x ∪ . . . ∪ N P ( S ) x t .Now, let 1 = τ ∈ Irr( B ( S )), and consider θ = τ × × · · · × ∈ Irr( B ( N )). Let χ ∈ Irr( B ( G )) over τ . Then χ has p ′ -degree, so χ N = θ . Then θ is invariant. Butthis is impossible unless t = 1. Thus we obtain that G = N P is almost simple, withsoc( G ) = N = S , and G/N a p -group.We are left with the case that X = G . Hence G/L is a direct product of non-abeliansimple groups of order divisible by p . Since L ⊆ N C G ( Q ) and G/N C G ( Q ) is a p -group,we have that G = N C G ( Q ). Write N = S z ×· · ·× S z u , where S is simple non-abelian, and { S z , . . . , S z u } are the different G -conjugates of S . Now, Q ∩ S ∈ Syl p ( S ). If 1 = x ∈ Q ∩ S and c ∈ C G ( Q ), then x c = x ∈ S . In particular, c ∈ N G ( S ), and therefore S E G .Hence S = N . By the Schreier theorem, we have that G/N is solvable, but this is notpossible. (cid:3)
As pointed out by the referee, the proof of Theorem 2.1 could be shortened somewhatby using the theory of the so called p ∗ -groups (see [26]).3. Almost Simple Groups
We now deal with almost simple groups with p -automorphisms. The first observationfollows from the classification of finite simple groups. We refer the reader to [7, § Lemma 3.1.
Let S be non-abelian simple with abelian Sylow p -subgroups and supposethat S has a non-trivial outer automorphism σ of p -power order. Then S is of Lie type GUNTER MALLE AND GABRIEL NAVARRO and either σ is a field automorphism and p does not divide the order of the group of outerdiagonal automorphisms of S , or one of the following holds: (1) p = 2 and S = PSL ( q ) with q ≡ , ; (2) p = 3 and S = PSL ( q ) with q ≡ , ; or (3) p = 3 and S = PSU ( q ) with q ≡ , .Proof. If S is sporadic or alternating, then Out( S ) has 2-power order, so p = 2. The onlysuch S with abelian Sylow 2-subgroups is J , but Out( J ) = 1.Thus, by the classification of finite simple groups, S is of Lie type. Here, if p = 2,then by the well-known classification by Walter [25], S is one of SL (2 f ) with f ≥ ( q ) with q ≡ , G (3 f +1 ) with f ≥
1. The groups not occurring inthe conclusion do not have outer automorphisms that are not field automorphisms. Nowassume that p >
2. Here σ is a product of diagonal, graph and field automorphisms. Non-trivial graph automorphisms of order p ≥ D ( q ), but hereSylow 3-subgroups are non-abelian (since, for example, PSL ( q ) is a section). Non-trivialouter diagonal automorphisms of order p ≥ A n and E .For S = E ( q ) or E ( q ) these have order 3, but Sylow 3-subgroups of S are non-abelian.For S = PSL n ( q ) or PSU n ( q ) the outer diagonal automorphisms have order dividing n ,but Sylow p -subgroups of S for 3 ≤ p | n are non-abelian, unless n = p = 3. The lattercases appear in our conclusion. (cid:3) We are thus reduced to considering groups of Lie type. For these, we require somenotions and results from Deligne–Lusztig theory. Let G be a simple algebraic group ofadjoint type and F : G → G a Steinberg endomorphism, with finite group of fixed points G = G F . Let ( G ∗ , F ) be in duality with G . Any field automorphism σ of G is inducedby a Frobenius map F : G → G commuting with F and the corresponding Frobeniusmap F : G ∗ → G ∗ induces a field automorphism σ ∗ of G ∗ = G ∗ F (see [24, § Lemma 3.2.
Let G be as above of adjoint Lie type with abelian Sylow p -subgroups for anon-defining prime p , and σ a field automorphism of G of p -power such that G h σ i hasnon-abelian Sylow p -subgroups. Then σ ∗ does not fix all classes of p -elements in G ∗ .Proof. First note that our assumptions on G force p >
2. Let F : G → G be a Frobeniusmap inducing σ and commuting with F . Since a Sylow p -subgroup P of G is abelian, itis contained in an F -stable maximal torus T of G (see [16, Thm. 25.14]), which we mayassume to be F -stable. In fact, if F is not very twisted, T contains a Sylow d -torus T d of G , where d is the order of q modulo p and q is the underlying field size of G , while inthe case of Suzuki and Ree groups, it contains a Sylow Φ-torus which again we denote T d , where Φ is the minimal polynomial over Q ( √
2) respectively Q ( √
3) of a primitive d throot of unity (see [4, 3F]). Since G and G ∗ , as well as T F and T ∗ F , have the same order, T ∗ F contains a Sylow p -subgroup of G ∗ , which is hence also abelian. Let σ ∗ be the dualautomorphism of G ∗ induced by F : G ∗ → G ∗ which we may and will assume to stabilise T ∗ . Let P ∗ be the Sylow p -subgroup of G ∗ contained in T ∗ F .By assumption, σ acts non-trivially on P . Thus the centraliser G := C G ( σ ) of σ , asubfield subgroup of G , does not contain a Sylow p -subgroup of G . Since σ ∗ is dual to σ , G ∗ := C G ∗ ( σ ∗ ) has the same order as G , so σ ∗ does also not centralise P ∗ . Now G ∗ -fusionin P ∗ is controlled by the relative Weyl group W = N G ∗ ( T ∗ d ) /C G ∗ ( T ∗ d ) of the Sylow torus RAUER’S HEIGHT ZERO CONJECTURE FOR PRINCIPAL BLOCKS 5 T ∗ d , by [14, Prop. 5.11]. Let q be the underlying field size of G F , so q = q p a for some a ≥
1. Since q and q have the same order d modulo p , the relative Weyl groups W in G ∗ and W = N G ∗ ( T ∗ d ) /C G ∗ ( T ∗ d ) in G ∗ agree, that is, σ ∗ commutes with the action of W on P ∗ . Now W has order prime to p as T ∗ Fd contains a Sylow p -subgroup of G ∗ F , while σ ∗ has some non-trivial orbit, of length divisible by p , on P ∗ . So σ ∗ induces additionalfusion on elements of P ∗ and our claim follows. (cid:3) Theorem 3.3.
Let p be a prime, S be a non-abelian simple group and S ≤ A ≤ Aut( S ) such that A/S is a p -group. Assume that all characters in the principal p -block of A havedegree prime to p . Then the Sylow p -subgroups of A are abelian.Proof. We will assume that A has non-abelian Sylow p -subgroups and argue by contra-diction. The case A = S is the main result of [12], so we may assume that A = S , S hasabelian Sylow p -subgroups, and we have to exhibit a character in the principal p -block of S that is not A -invariant.By Lemma 3.1 we may assume that S is of Lie type. We first discuss the groups showingup in Lemma 3.1(1)–(3). For S = PSL ( q ) the outer automorphism group is generated bydiagonal and field automorphisms. As q S does not have even order fieldautomorphisms. Now the principal 2-block of S contains two irreducible characters ofdegree ( q + ǫ ) /
2, where q ≡ ǫ (mod 4), ǫ ∈ {± } , that are not invariant under the diagonalautomorphism (see e.g. [10, Lemma 15.1]). For PSL ( q ) with 3 || ( q − q + 1)( q + q + 1) /
3, andsimilarly for PSU ( q ) with 3 || ( q + 1) the three characters of degree ( q − q − q + 1) / S does not have fieldautomorphisms of 3-power order. Thus, we may now assume that S is not one of theexceptions in Lemma 3.1.If p is the defining characteristic of S , then as S has abelian Sylow p -subgroups, wehave S = PSL ( q ) with q = p f > p . Here, all irreducible characters apart from theSteinberg character lie in the principal block. Let χ be a Deligne–Lusztig character ofdegree q − s generating the non-split maximaltorus T of order ( q +1) / (2 , q −
1) in the dual group. A field automorphism σ of S centralisesthe corresponding group over a subfield; since no proper subfield group contains a cyclicsubgroup of order ( q + 1) / (2 , q − σ must act non-trivially on T . Thus, χ is not σ -stable.So p is not the defining characteristic of S . Let G and G = G F be as above such that S = [ G, G ]. This is possible unless S is the Tits simple group, which has no non-trivialfield automorphisms. Any field automorphism of S extends to G . Now note that Sylow p -subgroups of G are also abelian, since G acts by diagonal automorphisms on S and p does not divide their order. Then by Lemma 3.2 there is some conjugacy class of a p -element s ∈ G ∗ not fixed by σ ∗ . By [24, Prop. 7.2] this implies that the correspondingLusztig series E ( G, s ) is not fixed by σ . In particular the semisimple characters in E ( G, s ),which lie in the principal p -block by [5, Thm.] are not fixed by σ . Since p does not dividethe order of the group of outer diagonal automorphisms of G , the number of charactersof S below G is not divisible by p , hence there is some character of S moved by σ , andwe conclude. (cid:3) Our main theorem now follows by combining Theorems 2.1 and 3.3.
GUNTER MALLE AND GABRIEL NAVARRO
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RAUER’S HEIGHT ZERO CONJECTURE FOR PRINCIPAL BLOCKS 7
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