# Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

aa r X i v : . [ m a t h . R T ] F e b BOUNDING p -BRAUER CHARACTERS IN FINITE GROUPSWITH TWO CONJUGACY CLASSES OF p -ELEMENTS NGUYEN NGOC HUNG, BENJAMIN SAMBALE, AND PHAM HUU TIEP

Dedicated to Burkhard K¨ulshammer on the occasion of his retirement.

Abstract.

Let k ( B ) and l ( B ) respectively denote the number of ordinary and p -Brauer irreducible characters in the principal block B of a ﬁnite group G . Weprove that, if k ( B ) − l ( B ) = 1, then l ( B ) ≥ p − p = 11 and l ( B ) = 9.This follows from a more general result that for every ﬁnite group G in which allnon-trivial p -elements are conjugate, l ( B ) ≥ p − p = 11 and G/ O p ′ ( G ) ∼ = C ⋊ SL(2 , G of order divisible by p , the numberof irreducible Brauer characters in the principal p -block of G is always at least2 √ p − − k p ( G ), where k p ( G ) is the number of conjugacy classes of p -elementsof G . This indeed is a consequence of the celebrated Alperin weight conjecture andknown results on bounding the number of p -regular classes in ﬁnite groups. Introduction

Let G be a ﬁnite group and p a prime. Bounding the number k ( G ) of conjugacyclasses of G and the number k p ′ ( G ) of p -regular conjugacy classes of G is a classicalproblem in group representation theory, one important reason being that k ( G ) isthe same as the number of non-similar irreducible complex representations of G and k p ′ ( G ) is the same as the number of non-similar irreducible representations of G overan algebraically closed ﬁeld F of characteristic p . It was shown recently in [HM,Theorem 1.1] that if G has order divisible by p , then k p ′ ( G ) ≥ √ p − − k p ( G ),where k p ( G ) denotes the number of conjugacy classes of p -elements of G . As it isobvious from the bound itself that equality could occur only when p − Mathematics Subject Classiﬁcation.

Primary 20C20, 20C33, 20D06.

Key words and phrases.

Finite groups, Brauer characters, conjugacy classes, Alperin weightconjecture.The second author is supported by the German Research Foundation (SA 2864/1-2 andSA 2864/3-1). The third author gratefully acknowledges the support of the NSF (grant DMS-1840702), the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at theInstitute for Advanced Study (Princeton).

Motivated by the study of blocks which contain a small number of characters,in this paper we focus on the extremal situation where G has a unique non-trivialconjugacy class of p -elements. Theorem 1.1.

Let p be a prime and G a ﬁnite group in which all non-trivial p -elements are conjugate. Then one of the following holds: (i) k p ′ ( G ) ≥ p . (ii) k p ′ ( G ) = p − and G ∼ = C p ⋊ C p − (Frobenius group). (iii) p = 11 , G ∼ = C ⋊ SL(2 , (Frobenius group) and k p ′ ( G ) = 9 . Finite groups with a unique non-trivial conjugacy class of p -elements arise naturallyfrom block theory. For a p -block B of a group G , as usual let Irr( B ) and IBr( B )respectively denote the set of irreducible ordinary characters of G associated to B and the set of irreducible Brauer characters of G associated to B , and set k ( B ) := | Irr( B ) | and l ( B ) := | IBr( B ) | . The diﬀerence k ( B ) − l ( B ) is one of the importantinvariants of the block B as it somewhat measures the complexity of B , and in fact,the study of blocks with small k ( B ) − l ( B ) has attracted considerable interest, see[KNST, KS, RSV] and references therein.It is well-known that k ( B ) − l ( B ) = 0 if and only if k ( B ) = l ( B ) = 1, in whichcase the defect group of B is trivial. What happens when k ( B ) − l ( B ) = 1? Brauer’sformula for k ( B ) (see [KNST, p. 7]) then implies that all non-trivial B -subsectionsare conjugate. (Recall that a B -subsection is a pair ( u, b u ) consisting of a p -element u ∈ G and a p -block b u of the centralizer C G ( u ) such that the induced block b Gu isexactly B .) Therefore, if B is the principal p -block of G and k ( B ) − l ( B ) = 1, thenall the non-trivial p -elements of G are conjugate.Given a p -block B of G , the well-known blockwise Alperin weight conjecture (BAW)claims that l ( B ) is equal to the number of G -conjugacy classes of p -weights of B (fordetails see Section 3). The conjecture implies l ( B ) ≥ l ( b ) where B and b arethe principal blocks of G and N G ( P ) respectively. It is easy to see that l ( b ) = k p ′ ( N G ( P ) / O p ′ ( N G ( P ))).Now suppose that k p ( G ) = 2. Then the main result of [KNST] asserts that, asidefrom very few exceptions, the Sylow p -subgroups of G are (elementary) abelian,and so let us assume for a moment that P ∈ Syl p ( G ) is abelian. It follows that N G ( P ) controls G -fusion in P , and thus N G ( P ) / O p ′ ( N G ( P )) has a unique non-trivialconjugacy class of p -elements as G does. Therefore, the BAW conjecture and (the p -solvable case of) Theorem 1.1 suggest the following, which we are able to proveusing only the known cyclic Sylow case of the conjecture. Theorem 1.2.

Let p be a prime and G a ﬁnite group in which all non-trivial p -elements are conjugate. Let B denote the principal p -block of G . Then one of thefollowing holds: (i) l ( B ) ≥ p . OUNDING p -BRAUER CHARACTERS 3 (ii) l ( B ) = p − and N G ( P ) / O p ′ ( N G ( P )) ∼ = C p ⋊ C p − (Frobenius group). (iii) p = 11 , G/ O p ′ ( G ) ∼ = C ⋊ SL(2 , (Frobenius group) and l ( B ) = 9 . Theorem 1.2 implies that if G is a ﬁnite group with k p ( G ) = 2 then k ( B ) ≥ p or p = 11 and k ( B ) = 10. Indeed, we obtain the following. Here, k ( B ) denotes thenumber of irreducible ordinary characters of height 0 in B . Theorem 1.3.

Let p be a prime and G a ﬁnite group in which all non-trivial p -elements are conjugate. Let B denote the principal p -block of G . Then k ( B ) ≥ p or p = 11 and k ( B ) = 10 . We mention another consequence, which is useful in the study of principal blockswith few characters, in particular the case k ( B ) − l ( B ) = 1. Note that by [KNST,Theorem 3.6], the Sylow p -subgroups of G then must be (elementary) abelian, andhence by [KM1], k ( B ) = k ( B ). Corollary 1.4.

Let p be a prime and G a ﬁnite group with principal p -block B . If k ( B ) − l ( B ) = 1 then k ( B ) = k ( B ) ≥ p or p = 11 and k ( B ) = 10 . For a quick example, let us assume that k ( B ) = 4 and l ( B ) = 3. Then Corol-lary 1.4 implies that p ≤

4, and since the case p = 3 is eliminated by [Lan, Corollary1.6], one ends up with p = 2, implying that the defect group of B must be of order4 by [Lan, Corollary 1.3], and thus is the Klein four group. This result was recentlyproved in [KS, § k ( B ) = l ( B ) + 1 = 5 and k ( B ) = l ( B ) + 1 = 7.)In Section 6 we go one step further and prove that k p ′ ( G ) ≥ ( p − / G with at most three classes of p -elements. As explained in Section 3, thisand the BAW conjecture then imply that l ( B ) ≥ ( p − / B ofgroups with 1 < k p ( G ) ≤

3. In general, we propose that l ( B ) ≥ √ p − − k p ( G )for arbitrary groups of order divisible by p , and this follows from [HM, Theorem1.1] and again the BAW conjecture. We should mention that our proposed boundcomplements the conjectural upper bound for the number l ( B ) proposed by Malleand Robinson [MR], namely l ( B ) ≤ p r ( B ) , where r ( B ) is the sectional p -rank of adefect group of B .The paper is organized as follows. In the next Section 2, we prove Theorem 1.1for p -solvable groups. In Section 3 we make a connection between Theorem 1.2 andother bounds on l ( B ) with the BAW conjecture. Section 4 reduces Theorem 1.2 toalmost simple groups of Lie type, which are then solved in Section 5. In Section 6we prove a general bound for the number of p -regular conjugacy classes in almostsimple groups without any assumption on the number of p -classes, and this will beused to achieve a right bound for k p ′ ( G ) for ﬁnite groups G with at most three classesof p -elements. Finally, the proof of Theorem 1.3 and more examples of applicationsof Theorem 1.2 are presented in Section 7. N. N. HUNG, B. SAMBALE, AND P. H. TIEP p -Solvable groups We begin by proving Theorem 1.1 for p -solvable groups. Theorem 2.1.

Let G be a p -solvable group with k p ( G ) = 2 . Then one of the followingholds: (i) k p ′ ( G ) ≥ p . (ii) k p ′ ( G ) = p − and G ∼ = C p ⋊ C p − (Frobenius group). (iii) p = 11 , G ∼ = C ⋊ SL(2 , (Frobenius group) and k p ′ ( G ) = 9 .Proof. We assume ﬁrst that O p ′ ( G ) = 1. Then P := O p ( G ) = 1 by the Hall–Higman lemma. Since every p -element is conjugate to an element of P , P must be aSylow p -subgroup. Since Z ( P ) ✂ G , it follows that P = Z ( P ) is elementary abelian.Moreover, C G ( P ) = P and G/P is a transitive linear group (on P ). We need toshow that k p ′ ( G ) = k p ′ ( G/P ) = k ( G/P ) ≥ p − G/P is a subgroup of the semilinear groupΓL(1 , p n ) ∼ = F × p n ⋊ Aut( F p n ) ∼ = C p n − ⋊ C n where P ∼ = F p n or one of ﬁnitely many exceptions. We start with the ﬁrst case.Since Aut( F p n ) ﬁxes some x ∈ P \ { } in the base ﬁeld F p , F × p n must be containedin G/P (otherwise

G/P cannot be transitive on P \ { } ). Now G/P has at least( p n − /n ≥ p − F × p n . The equality here occurs if andonly if n = 1, in which case G is the Frobenius group C p ⋊ C p − .Now suppose that G/P is one of the exceptions in Passman’s list (see [Sam1,Theorem 15.1] for detailed information). For p = 3 the claim reduces to | G/P | ≥ G ∼ = C ⋊ SL(2 ,

5) with p = 11 is the only exception.Finally, suppose that N := O p ′ ( G ) = 1. Since k p ( G ) = k p ( G/N ), the abovearguments apply to

G/N . Since at least one p -regular element lies in N \ { } , weobtain k p ′ ( G ) ≥ k p ′ ( G/N ) ≥ p unless p = 11 and G/N ∼ = C ⋊ SL(2 , k ′ ( G ) = 10.Then all non-trivial elements of N are conjugate in G . As before, N must be anelementary abelian q -group for some prime q = 11. Let N ≤ M ✂ G such that M/N ∼ = C . Then G/M acts transitively on the M -orbits of N \ { } . In particular,these M -orbits have the same size. Since the non-cyclic group M/N cannot act ﬁxedpoint freely on N , all M -orbits have size 1 or 11. In the second case, ( | N | − / | G/M | = 120. This leaves only the possibility that N is cyclic of order q ≥ G/ C G ( N ) is cyclic and we derive the contradiction G = G ′ N ≤ C G ( N ).It remains to deal with the case where M acts trivially on N . Here we may goover to G := G/ O ( G ) such that k ( G ) = k ′ ( G ) = 10. Since G acts transitively on N \ { } , we obtain that | N | − | G/N | = | G/M | = 120. Since G/ C G ( N ) ∈ OUNDING p -BRAUER CHARACTERS 5 { SL(2 , , A } , this leaves the possibilities | N | ∈ { , } . Now it can be checked bycomputer that there is no (perfect) group with these properties. (cid:3) Apart from ﬁnitely many exceptions, the proof actually shows that k p ′ ( G ) ≥ p n − n where | G | p = p n .The following result provides a bound for k p ′ ( G ) in p -solvable groups with threeconjugacy classes of p -elements. Theorem 2.2.

Let G be a p -solvable group with k p ( G ) = 3 . Then k p ′ ( G ) ≥ ( p − / with equality if and only if p > and G is the Frobenius group C p ⋊ C ( p − / .Proof. As in the proof of Theorem 2.1 we start by assuming O p ′ ( G ) = 1. Since theclaim is easy to show for p ≤

5, we may assume that p ≥ P := O p ( G ) = 1 and H := G/P . Suppose ﬁrst that | H | is divisible by p . Then k p ( H ) = 2 and k p ′ ( G ) = k p ′ ( H ) ≥ p − > p − H be a p ′ -group. Suppose that P possesses a characteristicsubgroup 1 < Q < P . Then P \ Q must be an H -orbit and therefore | P \ Q | isnot divisible by p . This is clearly impossible. Hence, P is elementary abelian and G ∼ = P ⋊ H is an aﬃne primitive permutation group of rank 3 (i. e. a point stabilizerhas three orbits on P ). These groups were classiﬁed by Liebeck [Lie].Let | P | = p n . Suppose ﬁrst that H ≤ ΓL(1 , p n ). Then H contains a semiregularnormal subgroup C ≤ F × p n . Clearly, C has exactly p n − | C | orbits on P \ { } each oflength | C | . Moreover, Aut( F p n ) ﬁxes one of these orbits and can merge at most n ofthe remaining. Hence, | C | + n | C | ≥ p n − | C | ≥ p n − n . Now there are at least p n − n + n conjugacy classes of H lying inside C . Since p n − p − ≥ p + . . . + p n − ≥ . . . + 2 n − = 2 n − ≥ n ( n + 1)2 , we obtain k p ′ ( H ) ≥ p − with equality if only if n = 1 and G ∼ = C p ⋊ C ( p − / .Now assume that H acts imprimitively on P = P × P interchanging P and P .Then K := N H ( P ) = N H ( P ) ✂ H and K/ C H ( P ) is a transitive linear group on P .Theorem 2.1 yields k ( K ) ≥ k ( K/ C H ( P )) ≥ p −

2. Since | H : K | = 2, the conjugacyclasses of K can only fuse in pairs in H . This leaves at least 1 + p − = p − conjugacyclasses of H inside K and there is at least one more class outside K . Altogether, k ( H ) ≥ p +12 .Next suppose that P = P ⊗ P considered as F q -spaces where q a = p n and H stabilizes P and P . Here | P | = q and | P | = q d ≥ q . By [Lie, Lemma 1.1], H hasan orbit of length ( q d − q d − q ), but this is impossible since H is a p ′ -group.The cases (A4)–(A11) in Liebeck [Lie] are not p -solvable. Cases (B) and (C) areﬁnitely many exception. Suppose that p = 7 and k ( H ) ≤

3. It is well-known that then

N. N. HUNG, B. SAMBALE, AND P. H. TIEP H ≤ S and therefore | P | ≤ n = 1 and G ∼ = C ⋊ C . Hence,let p ≥

11. From [Lie] we obtain | P | ≤ . Since the primitive permutation groups ofdegree at most 2 − p ≥

67. Thereare only three cases left, namely p ∈ { , , } and n = 2. Here A ≤ H/ Z ( H ).Since A is a maximal subgroup of PSL(2 , p ) (see [Hup, Hauptsatz II.8.27]), it followsthat H ∩ SL(2 , p ) = SL(2 , C := H/ SL(2 , ≤ GL(2 , p ) / SL(2 , p ) ∼ = C p − . Since H has an orbit of length at least ( p − /

2, we obtain 120 | C | = | H | ≥ ( p − / k ( H ) ≥ | C | > ( p − / p = 79 and | C | = 26. In this exception, H = SL(2 , . × C and obviously k ( H ) ≥ ·

13 = ( p − / N := O p ′ ( G ) = 1. Then the above arguments apply to G/N and we obtain k p ′ ( G ) ≥ k p ′ ( G/N ) > p − p -regular element lies in N . (cid:3) We remark that the p -solvability assumption in Theorem 2.2 will be removed inSection 6. 3. The blockwise Alperin weight conjecture

In this section, we will explain that, when the Sylow p -subgroups of G are cyclic,the main result Theorem 1.2 (and also Theorem 6.1) is a consequence of the knowncyclic Sylow case of the blockwise Alperin weight (BAW) conjecture and the p -solvableresults proved in the previous section.Let B be a p -block of G . Recall that l ( B ) denotes the number of irreducible Brauercharacters of B . A p -weight for B is a pair ( Q, λ ) of a p -subgroup Q of G and anirreducible p -defect zero character λ of N G ( Q ) /Q such that the lift of λ to N G ( Q )belongs to a block which induces the block B . The BAW conjecture claims that l ( B )is equal to the number of G -conjugacy classes of p -weights of B . In particular, theconjecture implies that l ( B ) ≥ l ( b ), where b is the Brauer correspondent of B (see[Alp, Consequence 1]). In fact, when a defect group of B is abelian, the conjectureis equivalent to l ( B ) = l ( b ) (see [Alp, Consequence 2]).Let P ∈ Syl p ( G ), and let B and b be respectively the principal blocks of G and N G ( P ). Assume that the BAW conjecture holds for ( G, p ). Since N G ( P ) is p -solvable, [Nav, Theorems 9.9 and 10.20] show that l ( B ) ≥ l ( b ) = k p ′ ( N G ( P ) / O p ′ ( N G ( P ))) = k ( N G ( P ) /P C G ( P )) . By Burnside’s fusion argument (see [Isa, Lemma 5.12]), H := N G ( P ) /P C G ( P )controls fusion in Z := Z ( P ). In particular, k p ( Z ⋊ H ) ≤ k p ( G ).Combining the above analysis with the results of the previous section, we de-duce that, if k p ( G ) = 2 then k p ( Z ⋊ H ) = 2 and l ( B ) ≥ p − p = 11 and OUNDING p -BRAUER CHARACTERS 7 N G ( P ) / O p ′ ( N G ( P )) ∼ = C ⋊ SL(2 , k p ( G ) = 3 then l ( B ) ≥ ( p − / k p ′ ( G ) ≥ ( p − /

2. Also, when k p ( G ) = 2, l ( B ) = p − k p ′ ( N G ( P ) / O p ′ ( N G ( P ))) = p −

1, which occurs if and only if N G ( P ) / O p ′ ( N G ( P )) isisomorphic to the Frobenius group C p ⋊ C p − , by Theorem 2.1.We therefore have the following, which was already mentioned in the introduction. Proposition 3.1.

Let p be a prime and G a ﬁnite group with k p ( G ) = 3 . Let B be the principal block of G . Then the blockwise Alperin weight Conjecture (for B )implies that l ( B ) ≥ ( p − / . Proposition 3.2.

Let p be a prime and G a ﬁnite group of order divisible by p . Let B be the principal block of G . Then the blockwise Alperin weight Conjecture (for B ) implies that l ( B ) ≥ √ p − − k p ( G ) .Proof. This follows from the above analysis and [HM, Theorem 1.1]. (cid:3)

We have seen that Theorem 1.2 holds for (

G, p ) if the BAW conjecture holds for(

G, p ). In particular, by Dade’s results [Dad] on blocks with cyclic defect groups, wehave proved the main results for groups with cyclic Sylow p -subgroups.We end this section by another consequence of the BAW conjecture on possiblevalues of k ( B ) and l ( B ) in blocks with k ( B ) − l ( B ) = 1. In the following theoremwe make use of Jordan’s totient function J : N → N deﬁned by J ( n ) := n Y p | n p − p where p runs through the prime divisors of n (compare with the deﬁnition of Euler’sfunction φ ). Theorem 3.3.

Let B be a p -block of a ﬁnite group G with defect d such that k ( B ) − l ( B ) = 1 . Suppose that B satisﬁes the Alperin weight Conjecture. Then one of thefollowing holds: (i) d = nk such that all prime divisors of n divide p k − . Moreover, if divides n ,then divides p k − . Here l ( B ) = X e | n p ek − ne J ( n/e ) . In particular, l ( B ) = p d − if n = 1 and l ( B ) > ( p k − φ ( n ) + p d − n if n > . (ii) p d l ( B ) 7 8 9 35 88 63 261 Conversely, all values for l ( B ) given in (i) and (ii) do occur in examples. N. N. HUNG, B. SAMBALE, AND P. H. TIEP

Proof.

By [HKKS, Theorem 7.1], B has an elementary abelian defect group D . Theequation k ( B ) − l ( B ) = 1 implies further that the inertial quotient E of B acts reg-ularly on D \ { } . It follows that all Sylow subgroups of E are cyclic or quaterniongroups. In particular, E has trivial Schur multiplier. Hence, the Alperin weightconjecture asserts that l ( B ) = k ( E ) (see [Sam1, Conjecture 2.6] for instance). Notethat D ⋊ E is a sharply 2-transitive group on D and those were classiﬁed by Zassen-haus [Zas] (see also [DM, Section 7.6]). Apart from the seven exceptions described in(ii), D ⋊ E arises from a Dickson near-ﬁeld F where ( F, +) ∼ = D and F × ∼ = E . Moreprecisely, there exists a factorization d = nk as in (i) such that F can be identiﬁedwith F q n where q = p k and the multiplication is modiﬁed as follows. Let F × q n = h ζ i .Let γ : F q n → F q n , x x q be the Frobenius automorphism of F q n with respectto F q . The hypotheses imply (with some eﬀort) that q has multiplicative order n modulo ( q − n . Hence, for every integer a there exists a unique integer a ∗ such that0 ≤ a ∗ < n and q a ∗ ≡ a ( q −

1) (mod ( q − n ) . It is easy to check that Γ : F × q n → h γ i , ζ a γ a ∗ is an epimorphism. We deﬁne F × := (cid:8) ( ζ a , γ a ∗ ) : 0 ≤ a < q n − (cid:9) ≤ F × q n ⋊ h γ i = ΓL(1 , q n ) . Note that F × is just the Singer cycle F × q if n = 1. Although diﬀerent choices for ζ may lead to non-isomorphic near-ﬁelds, the group F × is certainly uniquely deﬁned(as a subgroup of ( Z / ( q n − Z ) ⋊ ( Z /n Z ) for instance).It is easy to check that A := h ( ζ n , i = Ker Γ ✂ F × and F × /A ∼ = C n . This makesit possible to compute k ( E ) = k ( F × ) via Cliﬀord theory with respect to A . Thenatural actions of F × on A and on Irr( A ) are permutation isomorphic, by Brauer’spermutation lemma. Thus, instead of counting characters of A with a speciﬁc orderwe may just count elements. For a divisor e | n , let α ( e ) be the number of elementsin F × ∩ F q e which do not lie in any proper subﬁeld of F q e . Then β ( e ) := | F × ∩ F q e | = q e − e = X f | e α ( f ) . By M¨obius inversion we obtain α ( e ) = X f | e µ ( e/f ) q f − f . This is also the number of characters in Irr( A ) with inertial index e . These charactersdistribute into α ( e ) /e orbits under F × . Each such character has n/e distinct exten-sions to its inertial group and each such extension induces to an irreducible characterof F × . The number of character of F × obtained in this way is therefore α ( e ) n/e . In OUNDING p -BRAUER CHARACTERS 9 total, l ( B ) = k ( E ) = k ( F × ) = X e | n ne X f | e µ ( e/f ) q f − f . Now observe that n = P d | n J ( d ) for all n ≥

1. Hence, another M¨obius inversionyields X e | n ne X f | e µ ( e/f ) q f − f = X f | n q f − f n X e ′ | nf (cid:16) ne ′ f (cid:17) µ ( e ′ ) = X f | n q f − f n J ( n/f ) . If n >

1, then nφ ( n ) = n Q p | n p − p < J ( n ) and the second claim follows.Conversely, if d = nk satisﬁes the condition in (i), then a corresponding near-ﬁeld F can be constructed as above. This in turn leads to a sharply 2-transitivegroup G = F ⋊ F × . Now G has only one block B , namely the principal block, and l ( B ) = k ( F × ) is given as above. (cid:3) Reduction for Theorem 1.2

In this section we prove Theorem 1.2, assuming a result on bounding l ( B ) foralmost simple groups of Lie type that will be proved in Section 5. We restate Theo-rem 1.2 for the convenience of the reader. Theorem 4.1.

Let p be a prime and let G be a ﬁnite group with k p ( G ) = 2 . Let B be the principal p -block of G . Then l ( B ) ≥ p − or p = 11 and G/ O p ′ ( G ) ∼ = C ⋊ SL(2 , . Furthermore, l ( B ) = p − if and only if N G ( P ) / O p ′ ( N G ( P )) isisomorphic to the Frobenius group C p ⋊ C p − .Proof. Recall that B is isomorphic to the principal p -block of G/ O p ′ ( G ). We mayassume that O p ′ ( G ) = 1. Moreover, as the theorem is easy for p = 2, we assume p ≥

3. Also, since the case of cyclic Sylow follows from the blockwise Alperin weightconjecture, as explained in Section 3, we assume furthermore that P ∈ Syl p ( G ) is notcyclic. We aim to prove that l ( B ) > p − p = 11 and G ∼ = C ⋊ SL(2 , P is non-abelian. Then p ≤ p = 5, G is isomorphic to the sporadic simple Thompson group T h , and fromthe Atlas [Atl] we get l ( B ) = l ( B ( T h )) = 20 >

4, as desired. Let p = 3. Then S := O p ′ ( G ) is isomorphic to the Rudvalis group Ru , the Janko group J , the Titsgroup F (2) ′ , or the Ree groups F ( q ) with q = 2 b ± for b ∈ Z + , by [KNST] again.Since C G ( S ) ≤ O p ′ ( G ) = 1, G is almost simple. We now check with [GAP] that l ( B ( Ru )) = l ( B ( J )) = l ( B ( F (2) ′ )) = l ( B ( F (2))) = 9 > . Therefore we may assume that S = F ( q ) with q = 2 b ± for some b ∈ Z + and S ✂ G ≤ Aut( S ). By [Mal1, § §

7] (see also [Him, Table C5]), the principal3-block of F ( q ) ( q ≥

8) contains three irreducible Brauer characters (denoted by φ , φ , , and of course the trivial character) that are Aut( S )-invariant (since theirdegrees are unique in B ( S )), and thus we have l ( B ) ≥

3, as wanted.We may now assume that P is abelian. By Burnside’s fusion argument, all non-trivial p -elements of N G ( P ) are conjugate, i. e. N G ( P ) satisﬁes the hypothesis ofTheorem 2.1. Let N be a minimal normal subgroup of G . If N is elementary abelian,then N = P since every element of P is conjugate to some element of N . From O p ′ ( G ) = 1 it then follows that B is the only block of G . Hence, the theoremfollows from Theorem 2.1. Now let N = T × . . . × T n with non-abelian simplegroups T ∼ = . . . ∼ = T n . Since O p ′ ( G ) = 1, | T i | is divisible by p . Since non-trivial p -elements of the form ( x, , . . . ,

1) and ( x, x, , . . . ,

1) in N cannot be conjugate in G , we conclude that n = 1, i. e. N is simple. Since C G ( N ) ∩ N = Z ( N ) = 1 we have C G ( N ) ≤ O p ′ ( G ) = 1. Altogether, G ≤ Aut( N ), i. e. G is an almost simple group.Moreover, p ∤ | G/N | .Let N = A n be an alternating group. Recall that the Sylow p -subgroups of G (and N ) are not cyclic. Therefore, n ≥ p . But then the p -elements of cycle type ( p ) and( p, p ) are not conjugate in G . The sporadic and the Tits groups can be checked with[GAP] (or one appeals to Alperin’s weight conjecture proved in [Sam2]). Next let N be a simple group of Lie type in characteristic p . Then P can only be abelianif N ∼ = PSL(2 , p n ) for some n ≥ B , i. e. l ( B ) = l ( b ) where b is the principal block of N G ( P ). Now l ( b ) is the number of p -regular conjugacyclasses of the p -solvable group H := N G ( P ) / O p ′ ( N G ( P )). Hence, the claim followsfrom Theorem 2.1 unless possibly p = 11 and H = C ⋊ SL(2 , N ∼ = PSL(2 , ) and SL(2 ,

5) is not involved in N G ( P ).Finally, let N be a simple group of Lie type in characteristic diﬀerent from p . Insuch case, we show in Theorem 5.1 below that l ( B ) ≥ p , and thus the proof iscomplete. (cid:3) Principal blocks of almost simple groups of Lie type

We now prove the following result which is left oﬀ at the end of Section 4.

Theorem 5.1.

Let p ≥ be a prime and S = F (2) ′ a simple group of Lie type incharacteristic diﬀerent from p . Assume that the Sylow p -subgroups of S are abelianbut not cyclic. Let S ✂ G ≤ Aut( S ) such that p ∤ | G/S | . Let B be the principal p -block of G . Then l ( B ) ≥ p or G has at least two classes of non-trivial p -elements. We will work with the following setup. Let G be a simple algebraic group of simplyconnected type deﬁned over F q and F a Frobenius endomorphism on G such that and S = G / Z ( G ), where G := G F is the set of ﬁxed points of G under F . Let G ∗ be analgebraic group with a Frobenius endomorphism which, for simplicity, we denote bythe same F , such that ( G , F ) is in duality to ( G ∗ , F ). Set G ∗ := G ∗ F . As we will see OUNDING p -BRAUER CHARACTERS 11 below, the Brauer characters in the principal blocks of S and G arise from the so-called unipotent characters of G . These are the irreducible characters of G occurringin a Deligne–Lusztig character R GT (1), where T runs over the F -stable maximal toriof G , see [DM, Deﬁnition 13.19]. It is well-known that the unipotent characters of G all have Z ( G ) in their kernel, and so they are viewed as (unipotent) characters of S .From the assumption on P ∈ Syl p ( S ) and p , we may assume that S is not one ofthe types A , G , and B . Assume for a moment that S is also not a Ree groupof type F neither, so that F deﬁnes an F q -rational structure on G . Let d be themultiplicative order of q modulo p .By [KM2, Theorem A], which includes earlier results of Brou´e–Malle–Michel [BMM]and of Cabanes–Enguehard [CE], the p -blocks of G are parameterized by d -cuspidalpairs ( L , λ ) of a d -split Levi subgroup L of G and a d -cuspidal unipotent character λ of L F . In particular, the principal block of G corresponds to the pair consistingof the centralizer L d := C G ( S d ) of a Sylow d -torus S d of G and the trivial characterof L Fd . Moreover, the number of unipotent characters in B ( G ) is the same as thenumber of characters in the d -Harish-Chandra series associated to the pair ( L d , d -Harish-Chandra series are in one-to-one correspondence with the irreducible characters of the relative Weyl group of the d -cuspidal pair deﬁning the series. Therefore, the number of unipotent characters in B ( G ) is precisely the number of irreducible characters of the relative Weyl group W ( L d ) of L d .Assume that p ∤ | Z ( G ) | . Then, as the Sylow p -subgroups of S are abelian, those of G are abelian as well. In such situation, we follow [MM, § p -elements in G . In particular, by [Mal3, Proposition 2.2],we know that the order d of q modulo p deﬁned above is a unique positive integersuch that p | Φ d ( q ) with Φ d the d th cyclotomic polynomial dividing the generic orderof G . Furthermore, p is indeed a good prime for G (see [Mal3, Lemma 2.1]). LetΦ m d d be the precise power of Φ d dividing the generic order of G . Note that, by theassumption on Sylow p -subgroups of S and the main result of [KNST], a P ∈ Syl p ( S )must be elementary abelian, and thus Φ d ( q ) is divisible by p but not p . Therefore, P is isomorphic to the direct product of m d copies of C p . Since P is non-cyclic, m d > P to be inside the Sylow d -torus S d and let T d be an F -stable maximal torus of G containing S d , we deduce that the fusion of p -elements in P is controlled by therelative Weyl group W ( T d ) of T d . Therefore, the number of conjugacy classes of(non-trivial) p -elements of G , and hence of S , is at least | P | − | W ( T d ) | = p m d − | W ( T d ) | . Note that when d is regular for G , which means that C G ( S d ) is a maximal torusof G , the maximal torus T d can be chosen to be the same as L d = C G ( S d ), and thisindeed happens for all exceptional types and all d , except the single case of type E and d = 4 (see also [HSF, p. 18]).Recall that p ∤ | Z ( G ) | , and thus B ( G ) and B ( S ) are isomorphic, and, moreover, p is a good prime for G . By a result of Geck [Gec2, Theorem A], the restrictions ofunipotent characters of G in B ( G ) to p -regular elements form a basic set of Brauercharacters of B ( G ). In particular, l ( B ( S )) = l ( B ( G )) is precisely the number ofunipotent (ordinary) characters in B ( G ), which in turns is the number k ( W ( L d )) ofirreducible characters of W ( L d ), as mentioned above. Proposition 5.2.

Theorem 5.1 holds for groups of exceptional Lie types.Proof.

We will keep the notation above. In particular, G and G ∗ are ﬁnite reduc-tive groups of respectively simply-connected and adjoint type with S = G / Z ( G ) ∼ =[ G ∗ , G ∗ ]. First we note that the Sylow 3-subgroups of simple groups of type E or E are not abelian since their Weyl group (SO(5 , p ∤ | Z ( G ) | in all cases.We will follow the following strategy to prove the theorem for exceptional types.Let G be the extension of G ∗ to include ﬁeld automorphisms. Let H := h G ∩ G ∗ , C G ∩ G ( P ) i . Note that every unipotent character of S is G -invariant and extendible to its inertialsubgroup in Aut( S ), by results of Lusztig and Malle (see [Mal2, Theorems 2.4 and2.5]). In particular, every unipotent character in B ( S ) extends to a character in B ( G ). The result of Geck noted above then implies that each θ ∈ IBr( B ( S ))extends to some µ ∈ IBr( B ( G ∩ G )). Now µ H ∈ IBr( B ( H )). Moreover, as P C G ∩ G ( P ) ⊆ H , B ( G ∩ G ) is the only block of G ∩ G covering B ( H ) (see [RSV,Lemma 1.3]). It follows that µη ∈ IBr( B ( G ∩ G )) for every η ∈ IBr(( G ∩ G ) /H )by [Nav, Corollary 8.20 and Theorem 9.2], and thus(5.1) l ( B ( G ∩ G )) ≥ l ( B ( S )) | ( G ∩ G ) /H | = k ( W ( L d )) | ( G ∩ G ) /H | . Here we remark that ( G ∩ G ) /H is a quotient of ( G ∩ G ) / ( G ∩ G ∗ ) and thus cyclic.Also, the number l ( B ( G )) could be smaller than l ( B ( G ∩ G )), depending on howunipotent characters of S are fused under graph automorphisms, and this will beexamined below in a case by case analysis.Assume for now that d is regular for G (which means ( G , d ) = ( E , T d := L d as mentioned above. Recall that | P | = p m d and S then has atleast ( p m d − / | W ( L d ) | conjugacy classes of non-trivial p -elements. Assume that G has a unique class of non-trivial p -elements, and therefore we aim to prove that OUNDING p -BRAUER CHARACTERS 13 l ( B ( G )) ≥ p . Since C G ( P ) ﬁxes every class of p -elements of S , we deduce that(5.2) p m d − | W ( L d ) | ≤ | G ||h S, C G ( P ) i| ≤ d | G || H | ≤ dg | G ∩ G || H | , where d and g are respectively the orders of the groups of diagonal and graph auto-morphisms of S .We now go through various types of S to reach the conclusion, with the help of(5.1) and (5.2). For simplicity, set x := | ( G ∩ G ) /H | . The relative Weyl groups W ( L d ) for various types of G and d are available in [BMM, Table 3]. These relativeWeyl groups are always complex reﬂection groups and we will follow their notationin [BMM] as well as [Ben]. Recall that as the Sylow p -subgroups of S are non-cyclic,we may exclude the types B and G .Let S = G ( q ) with q >

2. Then d ∈ { , } , m = m = 2, and W ( L d ) isthe dihedral group D . Here all unipotent characters of S are Aut( S )-invariantunless q = 3 f for some odd f , in which case the graph automorphism fuses twocertain unipotent characters in the principal series, by a result of Lusztig (see [Mal2,Theorem 2.5]). In any case, the bound (5.1) yields l ( B ( G )) ≥ ( k ( D ) − x = 5 x .Together with (5.2), we have l ( B ( G )) ≥ x > √ x ≥ p p − > p − , as desired.For S = F ( q ) we have d ∈ { , , , , } with m = m = 4 and m = m = m =2. Here all unipotent characters of S are Aut( S )-invariant unless q = 2 f for someodd f , in which case the graph automorphism fuses eight pairs of certain unipotentcharacters. Also, W ( L , ) = G , W ( L , ) = G , and W ( L ) = G . In all cases wehave l ( B ) ≥ ( k ( W ( L d )) − x > (2 | W ( L d ) | x ) /m d ≥ ( p m d − /m d > p − . For all other exceptional types every unipotent character of S is Aut( S )-invariant,again by [Mal2, Theorem 2.5]. The bound (5.1) then implies that l ( B ( G ) ≥ k ( W ( L d )) x .On the other hand, the bound (5.2) yields dgx | W ( L d ) | ≥ p m d −

1. The routine esti-mates are then indeed suﬃcient to achieve the desired bound.As the arguments for D , E , E , E with d = 4, and E are fairly similar, weprovide details only for S = E ( q ) as an example. Then d ∈ { , , , , , , , , } with m , = 8, m , , = 4, and m , , , = 2. Going through various values of d , weobserve that k ( W ( L d )) m d > | W ( L d ) | for all relevant d . The above estimates thenimply that l ( B ( G )) m d ≥ k ( W ( L d )) m d x > | W ( L d ) | x ≥ p m d − , which in turns implies that l ( B ( G )) ≥ p .Assume that S = E ( q ) and d = 4. (Recall that d = 4 is not regular for type E .) Then m d = 2. By [BMM, Table 1], L d = S d .A , W ( L d ) = G and W ( T d ) is an extension of G by C for any maximal torus T d containing S d . Note that hereOut( S ) is the direct product of C gcd(2 ,q − and C f where q = ℓ f for some prime ℓ ,and thus is abelian. Let y := | G/ h S, C G ( P ) i| and arguing similarly as above, wehave l ( B ( G )) ≥ k ( W ( L d )) y = 16 y and p − ≤ y | W ( T d ) | = 768 y . If y ≥ l ( B ( G )) ≥ y ≥ y ≥ p −

1, as desired. If y = 1 then p ≤

23, and sincewe are done if p ≤

16, we may assume that p = 17 ,

19, or 23, but for these primes, p − | W ( T d ) | = 768, implying that S , and hence G , has more thanone class of p -elements. Lastly, if y = 2 then the only prime we need to take care ofis p = 37, but as 37 − | W ( T d ) | , S nowhas at least three classes of p -elements, implying that G has more than one class of p -elements, as desired.Finally, let S = F ( q ) with q = 2 n +1 ≥

8. Here the prime p divides exactly oneof Φ ( q ), Φ ( q ), Φ + ( q ) = q + √ q + 1, and Φ − ( q ) = q − √ q + 1, and m d = 2 inall cases. All the Sylow d -tori are maximal and their relative Weyl groups are D for d = 1, G for d = 2, and G for d = 4 ± . Now one just applies (5.1) and (5.2) toarrive at the desired bound. (cid:3) Proposition 5.3.

Theorem 5.1 holds for groups of classical types.Proof.

First consider S = PSL ǫ ( n, q ) with ǫ = ± and n ≥

3. Here, as usual,PSL + ( n, q ) := PSL( n, q ) and PSL − ( n, q ) := PSU( n, q ). Let e be the smallest positiveinteger such that p | ( q e − ǫ e ).Assume that p ∤ | Z (SL ǫ ( n, q )) | , and thus we may view P as a (Sylow) p -subgroupof SL ǫ ( n, q ). Since P is not cyclic, we have 2 e ≤ n . (If 2 e > n then P wouldbe contained in a torus of order q e − ǫ e , and hence cyclic.) Let α be an elementof F × q of order p . We then can ﬁnd an element x ∈ SL ǫ ( e, q ) of order p that isconjugate to diag( α, α ǫq , . . . , α ( ǫq ) e − ) over F q . Now we observe that the two elements x := diag( x , I n − e ) and y := diag( x , x , I n − e ) of SL ǫ ( n, q ) produce two correspondingelements of order p in S that cannot be conjugate in G , as desired.Now assume p | | Z (SL ǫ ( n, q )) | . As P ∈ Syl p ( S ) is abelian, this happens only when p = 3 (see [KS, Lemma 2.8]). The proof of [KNST, Lemma 2.5] shows that, in thiscase, S = PSL ǫ (3 , q ) with 3 | ( q − ǫ ) but 9 ∤ ( q − ǫ ). Moreover, q = ℓ f for someprime ℓ with 3 ∤ f , so the Sylow 3-subgroups of S (and G ) are elementary abelianof order 9. Suppose that l ( B ( G )) ≤

2. Then the irreducible Brauer characters in B ( G ) are 1 G , and possibly another character γ . On the other hand, it is known from[Gec1, Theorem 4.5] and [Kun, Table 1] that B ( S ) then contains precisely 5 distinctirreducible 3-Brauer characters, two of which, 1 S and α , are linear combinationsof the restrictions of the two unipotent characters of degrees 1 and q − ǫq to 3-regular elements, and thus are G -invariant; and three more β , β , β . It followsthat γ lies above α , but then none of the Brauer characters of B ( G ) can lie above β i , a contradiction. Hence l ( B ( G )) ≥

3, as required. (In fact, Brou´e’s abeliandefect group conjecture, and hence the blockwise Alperin weight conjecture, holds

OUNDING p -BRAUER CHARACTERS 15 for principal 3-blocks with elementary abelian defect groups of order 9, see [KK], andthus the bound l ( B ( G )) ≥ p is odd, we may view P ∈ Syl p ( S ) as a Sylow p -subgroup of Sp, SO, and GO. Let e be the smallest positiveinteger such that p | ( q e − e ≤ n by the non-cyclicity of P .Consider S = PSp(2 n, q ) with n ≥

2. Since SL(2 , q e ) < Sp(2 e, q ), we may ﬁnd anelement x in Sp(2 e, q ) of order p with spectrum { α, α q , . . . , α q e − , α − , α q , . . . , α − q e − } (see the proof of [NT, Proposition 2.6]). Note thatSp(2 e, q ) × Sp(2 e, q ) × Sp(2 n − e, q ) < Sp(2 e, q ) × Sp(2 n − e, q ) < Sp(2 n, q ) . Now one sees that the images of x := diag( x , I n − e ) and y := diag( x , x , I n − e ) in S are not conjugate in G .Consider S = Ω(2 n + 1 , q ) with q odd and n ≥

3. Since p | ( q e − λ ∈ {± } such that p | ( q e − λ ). Using the embedding C q e − λ ∼ = SO λ (2 , q e ) < GO λ (2 e, q ) , we may ﬁnd x ∈ GO λ (2 e, q ) of order p and with the spectrum { α ± , α ± q , . . . , α ± q e − } .This x then must be inside SO λ (2 e, q ) since it has order p . Note thatSO λ (2 e, q ) × SO λ (2 e, q ) × SO(2 n − e + 1 , q ) < SO(2 n + 1 , q ) . It follows that the images of x := diag( x , I n − e +1 ) and y := diag( x , x , I n − e +1 ) in S are of order p , and are not conjugate in G .For S = PΩ + (2 n, q ) with n ≥

4, using the same element x ∈ SO λ (2 e, q ) as in thecase of odd-dimensional orthogonal groups and the embeddingSO λ (2 e, q ) × SO λ (2 e, q ) × SO + (2 n − e, q ) < SO + (2 n, q ) , we arrive at the same conclusion.Finally, consider S = PΩ − (2 n, q ) with n ≥

4. If n = 2 e , then we have p | ( q n − p -subgroups of S are in fact cyclic, which is not the case.So n ≥ e + 1. As in the case of split orthogonal groups, but using the embeddingSO λ (2 e, q ) × SO λ (2 e, q ) × SO − (2 n − e, q ) < SO − (2 n, q ) , we have that G has at least two classes of non-trivial p -elements as well. This ﬁnishesthe proof. (cid:3) We have completed the proof of Theorem 5.1, and therefore the proof of Theo-rems 1.2 and 1.1 as well.6.

Groups with three p -classes In this section we prove the following result, which provides a bound for k p ′ ( G ) forgroups G with 3 conjugacy classes of p -elements. Theorem 6.1.

Let G be a ﬁnite group with k p ( G ) = 3 . Then k p ′ ( G ) ≥ ( p − / with equality if and only if p > and G is the Frobenius group C p ⋊ C ( p − / . We will prove that Theorem 6.1 follows from Theorem 2.2, [HM, Theorem 2.1]on bounding the number of orbits of p -regular classes of simple groups under theirautomorphism groups, the known cyclic Sylow case of the blockwise Alperin weightConjecture, and the following result. Theorem 6.2.

Let p be a prime and S a ﬁnite simple group with non-cyclic Sylow p -subgroups. Let S ✂ G ≤ Aut( S ) . Then k p ′ ( G ) ≥ p .Proof. The theorem is clear when p = 2 , | G | has at least 3 prime divisors.Therefore we may assume that p ≥

5. We also may assume that S is not a sporadicsimple group or the Tits group, as these could be checked directly using the charactertable library in [GAP].Let S = A n . Since the Sylow p -subgroups of S are not cyclic, we have n ≥ p ≥ A n has at least p − p . Thesecycles together with an involution of S produce at least p p -regular classes of G , asdesired.Next we assume that S is a simple group of Lie type in characteristic p . As before,one then can ﬁnd a simple algebraic group G of simply connected type deﬁned incharacteristic p and a Frobenius endomorphism F such that S = G / Z ( G ), where G = G F . According to [Car, Theorem 3.7.6], the number of semisimple classes of G is q r , where q is the size of the underlying ﬁeld of G and r is the rank of G . Therefore, k p ′ ( S ) ≥ k p ′ ( G ) k p ′ ( Z ( G )) ≥ q r | Z ( G ) | . To prove the theorem in this case, it suﬃces to prove that q r ≥ p | Z ( G ) || Out( S ) | .Using the known values of | Z ( G ) | and | Out( S ) | available in [Atl, p. xvi] for instance,it is straightforward to check the inequality for all S and relevant values of q, r and p , unless ( S, p ) is one of the following pairs { (PSL(2 , ) , , (PSL(3 , , , (PSL(3 , , , (PSU(3 , , , (PSU(3 , , } . Again the character tables of the corresponding almost simple groups are availablein [GAP] unless S = PSL(3 , |h x i| , | C S ( x ) | ) where x ∈ S is p -regular. Of course these elementscannot be conjugate in G .For the rest of the proof, we will assume that S is a simple group of Lie type incharacteristic ℓ = p and let G be a ﬁnite reductive group of adjoint type with socle S . (Note that G from now on is diﬀerent from before where it denotes the ﬁnitereductive group of simply-connected type.) Lemma 6.3.

Let

S, G and G as above. If k p ′ ( G ) ≥ p | Out( S ) | , then k p ′ ( G ) ≥ p . OUNDING p -BRAUER CHARACTERS 17 Proof.

Let Cl p ′ ( S ) denote the set of p -regular classes of S and n ( H, Cl p ′ ( S )) thenumber of orbits of the action of a group H on Cl p ′ ( S ). Let G := h G ∪ G i . Then k p ′ ( G ) ≥ n ( G, Cl p ′ ( S )) ≥ n ( G , Cl p ′ ( S )) = 1 | G | (cid:16) X c ∈ Cl p ′ ( S ) | Stab G ( c ) | (cid:17) ≥ | G | (cid:16) X c ∈ Cl p ′ ( S ) | Stab G ( c ) | (cid:17) = | G || G | n ( G , Cl p ′ ( S )) ≥ | G || G | k p ′ ( G ) | G /S | ≥ k p ′ ( G ) | Out( S ) | ≥ p, as claimed. (cid:3) Recall that p ≥

5. As the Sylow p -subgroups of S , where p is not the deﬁningcharacteristic of S , are non-cyclic, S is not one of the types A , B and G .1. Let G = PGL ǫ ( n, q ) with ǫ = ± , q = ℓ f and n ≥

3. Here as usual we use ǫ = +for linear groups and ǫ = − for unitary groups. Consider tori T i ( i ∈ { n − , n } ) of G of size ( q i − ( ǫ i ) / ( q − ǫ | T n − | , | T n | ) = 1, there exists t ∈ { n − , n } such that p ∤ | T t | . Note that the fusion of semisimple elements in T t is controlledby the relative Weyl group, say W t , of T t , which is the cyclic group of order t (see[MM, Proposition 5.5] and its proof for instance). Therefore, the number of p -regular(semisimple) classes of G with representatives in T t is at least q t − ( ǫ i t ( q − ǫ . Let k ∈ N be the order of q modulo p . Since the Sylow p -subgroups of S are notcyclic, we must have n ≥ k . Now one can check that q t − ( ǫ i t ( q − ǫ ≥ f gcd( n, q − ǫ p = | Out( S ) | p for all possible values of q, n and p . It follows that k p ′ ( G ) ≥ | Out( S ) | p, and therefore we are done in this case by Lemma 6.3.2. Let G = SO(2 n + 1 , q ) or PCSp(2 n, q ) for n ≥ q = ℓ f . Since p is odd, itdoes not divide both q n − q n + 1. Let T be a maximal torus of G of order either q n − q n +1 such that p ∤ | T | . The fusion of (semisimple) elements in T is controlledby its relative Weyl group, which is cyclic of order 2 n in this case. Therefore, thenumber of conjugacy classes with representatives in T is at least 1 + ( q n − / (2 n ),and it follows that k p ′ ( G ) ≥ q n − n , since S has at least one non-trivial unipotent class. Let k ∈ N be minimal such that p divides q k −

1. Since the Sylow p -subgroups of S are non-cyclic, we must have n ≥ k . Let n = 2. It then follows that k = 1 and, as p ≥

5, we have q ≥

9, and thus the desired inequality 2+( q − / ≥ f p = p | Out( S ) | follows easily. So let n ≥

3, and hence Out( S ) is cyclic of order f gcd(2 , q − q n − n ≥ f gcd(2 , q − p for all the relevant values of p, q and n , unless ( n, p, q ) = (4 , , k p ′ ( G ) ≥ p | Out( S ) | , and the theorem follows by Lemma 6.3.3. Let G = PCO ǫ (2 n, q ) with ǫ = ± , q = ℓ f and n ≥

4. Here | Out( S ) | =2 f gcd(4 , q n − ǫ

1) unless ( n, ǫ ) = (4 , +), in which case | Out( S ) | = 6 f gcd(4 , q n − ǫ n ≥ k where k is minimal such that p | ( q k − k ∤ n . A maximal torus of G of size q n − ǫ q n − ǫ np -regular classes, which are suﬃcient for the desired bound of p | Out( S ) | unless( n, ǫ, q, p ) = (4 , + , , , ± , , S has elements of at least 5 diﬀerent orders coprime to 5, which makes k p ′ ( G ) ≥ p .Now we may assume k | n . The case ǫ = − and p | ( q k −

1) can be treated as aboveusing a torus of size q n + 1, with a note that gcd( q n + 1 , q k − | p . So we assume that ǫ = + or ǫ = − and p | ( q k + 1).Observe that now G has tori of size q n − ± p -regular elements.Also, the non-trivial conjugacy classes with representatives in these two tori haveonly one possible common class, which is an involution class. Therefore, k p ′ ( G ) ≥ q n − − n −

1) + q n − − n −

1) = 2 + q n − − n − h ( q, n ) . It turns out that the desired bound h ( q, n ) ≥ p | Out( S ) | is satisﬁed unless ( S, p ) =( P Ω − (2) , P Ω +12 (2) , P Ω +12 (3) , P Ω +16 (2) , P Ω +16 (3) , P Ω +8 (4) , P Ω +8 ( q ) , p ) with q ≤

29 and p | ( q + 1) but p ∤ ( q − S, p ) = ( P Ω − (2) , P Ω +12 (2) , P Ω +8 (4) , | S | . For ( S, p ) = ( P Ω +16 (2) , ±

1, we observe that G = S has at least 126 /

14 = 9 classes of 127-elements, 42 /

14 = 3 classes of 43-elements, and 84 /

14 = 6 classes of 129-elements, which implies that G has at least10 classes of { , , } -elements, and hence, by also including classes of elements OUNDING p -BRAUER CHARACTERS 19 of order 1,2, 3, 5, 7, 9, 13, we have the claimed bound. The same strategy also worksfor ( P Ω +12 (3) ,

13) and ( P Ω +16 (3) , S, p ) = ( P Ω +8 ( q ) , p ) with q ≤

29 and p | ( q + 1)but p ∤ ( q − S )-orbits on p -regularclasses of S in the proof of [HM, Lemma 4.6], we end up with the open cases( q, p ) ∈ { (4 , , (5 , , (8 , , (9 , , (11 , } . These groups can be realized as permutation groups (of degree 21 , ,

888 in the lastcase) in [GAP]. In each case we constructed enough random p -regular elements in S and computed their centralizer orders to make sure that these elements are notconjugate in G .4. Now we turn to the case where S is of exceptional type diﬀerent from B and G . To conveniently write the order | S | and its factors, we use Φ d to denote the d thcyclotomic polynomial over the rational numbers.As above let G be a ﬁnite reductive group (over a ﬁeld of size q ) of adjoint typewith socle S , and assume that | G | = q N Q i Φ i ( q ) a ( i ) for suitable positive integers a ( i )and N . (Indeed, N is the number of positive roots in the root system correspondingto S .) First we assume that the Sylow p -subgroups of G are not abelian. Then p must divide the order of the Weyl group of G , and thus G is one of the types E , E , E , and E and p ≤

7. Using elementary number theory, one now easily observesthat | S | has at least 8 diﬀerent prime divisors, and hence k p ′ ( G ) ≥ ≥ p . So wemay and will assume that the Sylow p -subgroups of G (and S ) are abelian (but notcyclic). It follows that there exists a unique d ∈ N such that p | Φ d ( q ) and a ( d ) ≥ G = G ( q ) with q = ℓ f ≥

3. We then have p | ( q − G = S hasmaximal tori of coprime orders Φ ( q ) and Φ ( q ), which are furthermore coprime to p since p ≥

5. The relative Weyl groups of these tori have order 6 (see [BMM, Tables1 and 3] for sizes of Weyl groups of various maximal tori). Therefore, k p ′ ( G ) ≥ ( q ) −

16 + Φ ( q ) −

16 = 1 + q . It is now suﬃcient to check that 1 + q / ≥ f p , but this is straightforward. The case S = F ( q ) is handled similarly by considering two maximal tori of orders Φ ( q ) andΦ ( q ).Let S = F ( q ) with q = 2 m +1 ≥

8. Then we have p | Φ ( q )Φ ( q )Φ ( q ). Usingmaximal tori of orders Φ ± ( q ) := q ± p q + q ± √ q + 1 with the relative Weylgroup of order 12, we end up with k p ′ ( G ) ≥ +12 ( q ) −

112 + Φ − ( q ) −

112 = 1 + q + q . Certainly 1 + ( q + q ) / ≥ (2 m + 1) p for all possible values of p and m except( p, m ) = (13 , Let S = D ( q ). We then have p | Φ ( q )Φ ( q )Φ ( q )Φ ( q ). The relative Weyl groupof a maximal torus of order Φ ( q ) has order 4, and thus the number of classes withrepresentatives in this torus is at least 1 + ( q − q ) /

4, which in turn is at least3 f p , as we wanted, unless ( q, p ) = (2 , , , S, p ) = ( D (4) , S has at least (4 − ) / (4) = 241 and thus,as Out( S ) is cyclic of order 6, we are done unless G = Aut( S ). In fact, even for G = Aut( S ), one just notices that G has at least 60 / S, p ) = ( D (3) , S has at least 18classes of elements of order Φ (3) = 73, which produces at least 6 classes for G .Now note that SL(2 , ≤ S , and by using [Atl], we then observe that SL(2 , S , has elements of orders 1, 2, 3, 4, 6, 7, 14, which produce 7 more 13-regularclasses, as wanted.For G = E ( q ) ad , we have p | Φ d ( q ) for some d ∈ { , , , , } . Consider the(semisimple) classes with representatives in a maximal torus of size Φ ( q ), with noticethat this torus has the relative Weyl group of order 9, we obtain k p ′ ( G ) ≥ q + q , which is certainly at least | Out( S ) | p for every relevant q and p . Similar argumentsalso work for G = E ( q ) ad and E ( q ) ad , but using a maximal torus of respectivelysize Φ ( q ) and Φ ( q )Φ ( q ) or Φ ( q )Φ ( q ), depending on which size is coprime to p .For G = E ( q ) with q = ℓ f we have p | Φ d ( q ) for some d ∈ { , , , , , , , , } and G has a maximal torus of size Φ ( q ) with the relative Weyl group of order 30,and it follows that k p ′ ( G ) ≥ ( q ) − ≥ f p, as desired. This concludes the proof of Theorem 6.2. (cid:3) We are now in the position to prove the main result of this section.

Proof of Theorem 6.1.

Assume that the theorem is false and let G be a minimalcounterexample. In particular, G is not isomorphic to the Frobenius group F p := C p ⋊ C ( p − / and k p ′ ( G ) ≤ ( p − /

2. Since k p ( G/ O p ′ ( G )) = k p ( G ) = 3 and k p ′ ( G/ O p ′ ( G )) ≤ k p ′ ( G ), we have O p ′ ( G ) = 1 or G/ O p ′ ( G ) ∼ = F p . In the lattercase G has a cyclic Sylow p -subgroup and hence cannot be a counterexample, asshown in Section 3, and thus we have O p ′ ( G ) = 1. Let N be a minimal normal sub-group of G . It follows that p | | N | , and hence k p ( G/N ) < k p ( G ). Now since k p ( G/N )cannot be 2 by Theorem 1.2, we must have p ∤ | G/N | and moreover, N is the uniqueminimal normal subgroup of G . OUNDING p -BRAUER CHARACTERS 21 We are done if N is abelian by Theorem 2.2. So we may assume that N is a directproduct of, say n , copies of a non-abelian simple group, say S . Note that p | | S | since p | | N | . Therefore, the assumption k p ( G ) = 3 implies that n ≤ n = 2. Let m ( S, p ) be the number of Aut( S )-orbits on p -regularclasses of S . We then have k p ′ ( G ) ≥ m ( S, p )( m ( S, p ) + 1) . It was shown in [HM, Theorem 2.1] that either m ( S, p ) > √ p − S, p ) belongsto a list of possible exceptions described in [HM, Table 1]. For the former case, wehave k p ′ ( G ) > √ p − √ p − > p − , which is a contradiction. For the latter case, going through the list of exceptions, wein fact still have m ( S, p )( m ( S, p ) + 1)2 > p − , which again leads to a contradiction.Finally we may assume that n = 1, which means that G is an almost simple groupwith socle S . Furthermore, p ∤ | G/S | . The theorem now follows from Section 3 whenSylow p -subgroups of S are cyclic and from Theorem 6.2 otherwise. This completesthe proof. (cid:3) Theorem 1.3 and further applications

We now derive Theorem 1.3, which is restated, from Theorem 1.2.

Theorem 7.1.

Let p be a prime and G a ﬁnite group in which all non-trivial p -elements are conjugate. Let B denote the principal p -block of G Then k ( B ) ≥ p or p = 11 and k ( B ) = 10 .Proof. The theorem follows from Theorem 1.2 and [KM1] when the Sylow p -subgroupsof G are abelian. Assume otherwise. Then, as mentioned before, by [KNST, Theorem1.1], either(a) p = 3 and O p ′ ( G/ O p ′ ( G )) is isomorphic to Ru , J or F ( q ) ′ with q = 2 b ± for anonnegative integer b , or(b) p = 5 and G/ O p ′ ( G ) is isomorphic to T h .We now just proceed as in the proof of Theorem 4.1, but with height 0 charactersinstead of Brauer characters. For (b) we have k ( B ) = k ( B ( T h )) = 20 >

5, andwe are done. For (a) we may assume that G is almost simple. As k ( B ( Ru )) = k ( B ( J )) = k ( B ( F (2) ′ )) = k ( B ( F (2))) = 9by [GAP], we may now assume that S = F ( q ) ′ with q = 2 b ± for some b ∈ Z + and S ✂ G ≤ Aut( S ). According to [Mal1, § § S contains the Steinberg character denoted by χ (of degree q ), the semisimplecharacter denoted by χ , (of degree ( q − q + 1) ( q − q + 1)), and the trivialcharacter, all of which are 3 ′ -degree and Aut( S )-invariant, implying that k ( B ) ≥ (cid:3) Finally, we provide some more examples of applications of Theorem 1.2 in thestudy of principal blocks with few characters.

Theorem 7.2.

Let G be a ﬁnite group with a Sylow p -subgroup P and the principal p -block B . Assume that k ( B ) = 5 and l ( B ) = 4 . Then P ∼ = C .Proof. By Theorem 1.2, we have p ≤

5. By [KNST, Theorem 3.6], P is (elementary)abelian. It then follows by [KM1] that the ordinary irreducible characters in B allhave p ′ -degree, and thus k ( B ) = 5. However, by [Lan, Corollaries 1.3 and 1.6],5 = k p ′ ( B ) is divisible by p if p = 2 or 3, which cannot happen.So we are left with p = 5. The equality part of Theorem 1.2 then implies that N G ( P ) / O p ′ ( N G ( P )) is isomorphic to the Frobenius group C p ⋊ C p − . In particular, P ∼ = C , as wanted. (cid:3) Theorem 7.3.

Let G be a ﬁnite group with a Sylow p -subgroup P and the principal p -block B . Assume that k ( B ) = l ( B ) + 1 = 7 . Then P ∼ = C .Proof. Again by Theorem 1.2, we have p ≤ p = 2 or 3do not occur by [Lan, Corollaries 1.3 and 1.6]. If p = 7 then the equality part ofTheorem 1.2 implies that N G ( P ) / O p ′ ( N G ( P )) ∼ = C ⋊ C , yielding that P ∼ = C , asclaimed.We now eliminate the possibility p = 5. Assume so. By Theorem 3.3, G is not p -solvable and has a non-cyclic Sylow p -subgroup. As in the proof of Theorem 4.1,we may assume that G is an almost simple group with a socle S of Lie type incharacteristic not equal to p and p ∤ | G/S | . Moreover, P is abelian but non-cylic.The proof of Proposition 5.3 then shows that, when S is of classical type, G hasmore than one class of non-trivial p -elements, contradicting the assumption that k ( B ) − l ( B ) = 1. Also, the proof of Proposition 5.2 shows that l ( B ) ≥ S is of exceptional types except possibly type G . (Indeed, the principal block of G ( q ) has exactly 6 irreducible modular characters when p | Φ , ( q ) = q ±

1, since k ( W ( L , )) = k ( D ) = 6.) So assume S = G ( q ). Note that, since P is not cyclic,5 = p | ( q ±

1) and hence q is not an odd power of 3, implying that every unipotentcharacter of S (including 6 in B ( S )) is Aut( S )-invariant, by [Mal2, Theorem 2.5].However, a quick inspection of the principal block of G ( q ) (see [His, Theorems Aand B]) reveals that it contains two (families of) non-unipotent characters of diﬀerentdegrees, implying that B ( G ) contains at least 2 irreducible ordinary characters lyingover non-unipotent characters of S . It follows that k ( B ( G )) ≥ (cid:3) OUNDING p -BRAUER CHARACTERS 23 We conclude by noting that, while Theorem 7.2 can also be deduced from the mainresult of [RSV] on principal blocks with exactly 5 irreducible ordinary characters,Theorem 7.3 is new.

Acknowledgments

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Abh. Math. Sem. Univ. Hamburg (1935),187–220. OUNDING p -BRAUER CHARACTERS 25 Department of Mathematics, The University of Akron, Akron, OH 44325, USA

Email address : [email protected] Institut f¨ur Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Univer-sit¨at Hannover, Welfengarten 1, 30167 Hannover, Germany

Email address : [email protected] Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

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