aa r X i v : . [ m a t h . R T ] J u l Character tables and defect groups
Benjamin Sambale ∗ July 10, 2020
Abstract
Let B be a block of a finite group G with defect group D . We prove that the exponent of the centerof D is determined by the character table of G . In particular, we show that D is cyclic if and onlyif B contains a “large” family of irreducible p -conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small caseseven the isomorphism type of D is determined in this situation. Moreover, it can read off from thecharacter table whether | D/D ′ | = 4 where D ′ denotes the commutator subgroup of D . We alsopropose a new characterization of nilpotent blocks in terms of the character table. Keywords: character table, defect groups
AMS classification:
A major problem in character theory is to decide which properties of a finite group G can be read offfrom the complex character table X ( G ) of G . In this note we focus on properties of p -blocks of G andtheir defect groups. For motivational purpose we review some results on the principal p -block of G (orany block of maximal defect). It is known that X ( G ) determines the following properties of a Sylow p -subgroup P of G :(1) | P | (only the first column of X ( G ) is needed).(2) whether P is abelian. For p = 2 , this is an elementary result of Camina–Herzog [5] (cf. [27]), butit requires the classification of finite simple groups (CFSG for short) if p is odd (see [17, 26]). If P is abelian, also the isomorphism type of P can be read off from X ( G ) , albeit there is no easy wayof doing this (see [17]).(3) the exponent of the center Z( P ) (see [24, Corollary 3.12]).(4) whether P E G (in fact, all normal subgroup orders).(5) whether P has a normal p -complement, i. e. whether the principal block is nilpotent (only the firstcolumn of X ( G ) is needed, see [24, Theorem 7.4]).(6) whether N G ( P ) = P C G ( P ) , i. e. whether the principal block has inertial index . This was done byNavarro–Tiep–Vallejo [30, Theorem D] for p > and by Schaeffer-Fry–Taylor [39, Theorem 1.7] if p = 2 . Both cases rely on the CFSG. ∗ Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1,30167 Hannover, Germany, [email protected]
17) whether N G ( P ) = P (see Navarro–Tiep–Turull [29] for p > and Schaeffer-Fry [38] for p = 2 ).Again the CFSG is required.(8) whether P is a TI set. The case p = 2 appeared in Chillag–Herzog [6, Corollary 7] and the authorhas verified the result for p > via the CFSG.(9) the exponent of the abelianization P/P ′ if p = 2 . This is a special case of a conjecture by Navarro–Tiep [28] proved by Malle [20] using the CFSG.(10) whether | P/P ′ | = 4 (see [25]).The results on the exponents of Z( P ) and P/P ′ are of interest, because X ( G ) does not determine exp( P ) (consider the non-abelian groups of order p where p > ).Now let B be an arbitrary p -block of G with defect group D . The distribution of irreducible charactersinto p -blocks is given by X ( G ) (see [23, Theorem 3.19]) and the order of D can be computed by theformula | D | = max n | G | p χ (1) p : χ ∈ Irr( B ) o (here and in the following, n p and n p ′ denote the p -part and p ′ -part of an integer n ). An element g ∈ G is conjugate to an element of D if and only if χ ( g ) = 0 for some χ ∈ Irr( B ) (see [12, Lemma 22]). Inparticular, we can decide if D E G . Whether or not we can determine if D is abelian would follow fromthe still unproven Height Zero Conjecture of Richard Brauer. Recently, Gabriel Navarro has asked meif X ( G ) determines if D is cyclic. As far as we know this has not yet been observed in the literature(an explicit conjecture for p ≤ appeared in [36]). We give an affirmative answer in terms of Galoistheory. Recall that χ, ψ ∈ Irr( G ) are called p -conjugate if there exists a Galois automorphism γ of Q such that χ γ = ψ and γ ( ζ ) = ζ for all p ′ -roots of unity ζ (see next section). Theorem 1.
Let B be a p -block of a finite group G with defect d > . Then B has cyclic defect groupsif and only if Irr( B ) contains a family of p -conjugate characters of size divisible by p d − . Next we show that (3) above generalizes to blocks. Although this implies Theorem 1, there is apparentlyno simple formula to compute exp(Z( D )) from X ( G ) . Theorem 2.
Let B be a block of a finite group G with defect group D . Then the exponent of the center Z( D ) is determined by the character table of G . If D is known to be abelian, an explicit formula for exp( D ) can be given in terms of the field of values Q ( B ) := Q ( χ ( g ) : χ ∈ Irr( B ) , g ∈ G ) . For a positive integer n we denote the n -th cyclotomic field by Q n . Theorem 3.
Let B be a p -block of a finite group G with abelian defect group D = 1 . Let m := | G | p ′ .Then exp( D ) = p | Q ( B ) : Q ( B ) ∩ Q m | p . If | D | ≤ p , then even the isomorphism type of D is determined by the character table. Our last result is a block-wise version of (10). 2 heorem 4.
Let B be a -block of a finite group G with defect group D of order d ≥ . Then | D/D ′ | = 4 if and only if | Irr( B ) | < d and Q ( B ) Q | G | ′ ∩ Q d = Q ( ζ ± ζ − ) where ζ ∈ C is a primitive d − -th root of unity. In particular, the character table of G determines if B has tame representation type. Recall that a block B with abelian defect group is nilpotent if and only if B has inertial index . By workof Okuyama–Tsushima [31, Proposition 1 and Theorem 3], B is nilpotent with abelian defect group ifand only if all characters in Irr( B ) have the same degree. More generally, it has been conjectured (andverified in many cases) by Malle–Navarro [22] that B is nilpotent if and only if all height zero charactershave the same degree (an invariant of X ( G ) ). A different characterization of nilpotent blocks in termsof the focal subgroup was proved by Kessar–Linckelmann–Navarro [16]. It is however not clear if the(order of the) focal subgroup is encoded in X ( G ) . The same remark applies to another conjecturalcharacterization by Puig in terms of counting Brauer characters in Brauer correspondents (see [8,Conjecture 6.3.3] and [40]). In the last section of this paper we propose a strengthening of Puig’sConjecture characterizing nilpotent blocks by a single invariant which is derived from lower defectgroups and can be computed from X ( G ) (see Conjecture 8).Note that (7) above does not admit a direct analog for non-principal blocks by Brauer’s third maintheorem (when N G ( P ) is replaced by the inertial group of a Brauer correspondent). Our notation is fairly standard and follows [23]. As usual, we set k ( B ) := | Irr( B ) | and l ( B ) := | IBr( B ) | for every block B of a finite group G . The generalized decomposition matrix Q = ( d xχϕ ) of B has size k ( B ) × k ( B ) and entries in the cyclotomic field Q exp( D ) where D is a defect group of B . The rows of Q are indexed by χ ∈ Irr( B ) and the columns are indexed by pairs ( x, ϕ ) where x ∈ D and ϕ ∈ IBr( b ) forsome Brauer correspondent b of B in C G ( x ) . Let G be the Galois group of Q | G | with fixed field Q | G | p ′ .Characters in the same G -orbit are called p -conjugate. Characters fixed by G are called p -rational. Wemake use of the natural isomorphisms G ∼ = Gal( Q | G | p | Q ) ∼ = ( Z / | G | p Z ) × . In this way G acts on the rows and columns of Q via d xχ γ ,ϕ = γ ( d xχϕ ) = d x γ χϕ ( γ ∈ G ) . We recall that the characters of a nilpotent block B with defect group D were parameterized byBroué–Puig [4] using the so-called ∗ -construction. More precisely, there exists a p -rational character χ ∈ Irr( B ) of height such that Irr( B ) = { λ ∗ χ : λ ∈ Irr( D ) } . For γ ∈ G , we have ( λ ∗ χ ) γ = ( λ γ ) ∗ χ . Proof of Theorem 1.
Let D be a defect group of B . If D is not cyclic, then the generalized decom-position matrix Q has entries in Q p d − . Hence, the lengths of the G -orbits on the rows of Q divide ϕ ( p d − ) = p d − ( p − . So there is no family of p -conjugate characters in Irr( B ) of size divisible by p d − .Now suppose that D = h x i is cyclic. If p = 2 , then B is nilpotent, because the inertial index of B is .By Broué–Puig [4], we have Irr( B ) = { λ ∗ χ : λ ∈ Irr( D ) } for some -rational χ . Let λ , . . . , λ d − be the3aithful characters of Irr( D ) . Then λ ∗ χ, . . . , λ d − ∗ χ is a family of -conjugate characters of B of size d − . Finally let p > . Then G ∼ = ( Z / | G | p Z ) × is cyclic and the rows and columns of Q form isomorphic G -sets by Brauer’s permutation lemma (see [9, Lemma IV.6.10]). Let b be a Brauer correspondent of B in C G ( D ) . For u ∈ D \ { } the Brauer correspondents b u := b C G ( u ) are nilpotent. In particular, l ( b u ) = 1 and every such u labels a unique column of Q . Two elements u, v ∈ D determine the samecolumn of Q if and only if they are conjugate under the inertial quotient N := N G ( D, b ) / C G ( D ) . Weregard N as a p ′ -subgroup of Aut( D ) . Since all generators of D are conjugate under G , the G -orbit ofthe column of Q labeled by x has size | Aut( D ) : N | ≡ p d − ) . The corresponding orbit on therows of Q yields the desired family of p -conjugate characters of Irr( B ) We remind the reader that every x ∈ G can be written uniquely as x = x p x p ′ = x p ′ x p where the p -factor x p is a p -element and the p ′ -factor x p ′ is a p ′ -element. The p -section of x is the set of elements y ∈ G such that x p and y p are conjugate.In the following we work over a “large enough“ complete discrete valuation ring O such that theresidue field O / J( O ) is algebraically closed of characteristic p . The remaining theorems are based onthe following observation. Proposition 5.
Let B be a p -block of G with defect group D . For a given p -element x ∈ G , thecharacter table determines the number of Brauer correspondents of B in C G ( x ) with defect group D .Proof. We assume that the column of the character table X = X ( G ) corresponding to x is given. Let q = p be another prime. By [24, Theorem 7.16], we find all elements g ∈ G such that the q ′ -factorof g is conjugate to x . By induction on the number of prime divisors of the order of an element, thewhole p -section S of x can be spotted in X . Let y , . . . , y l ∈ C G ( x ) be representatives for the conjugacyclasses of p -regular elements in C G ( x ) . Then the elements xy , . . . , xy l represent the conjugacy classesinside S (see [23, p. 105]). Let IBr(C G ( x )) = { ϕ , . . . , ϕ l } . We construct the matrices X x := (cid:0) χ ( xy i ) : χ ∈ Irr( B ) , ≤ i ≤ l (cid:1) ,Q x := (cid:0) d xχϕ i : χ ∈ Irr( B ) , ≤ i ≤ l (cid:1) , (2.1) Y x := (cid:0) ϕ i ( y j ) : 1 ≤ i, j ≤ l (cid:1) and observe that X x = Q x Y x can be read off of X (see [23, Corollary 5.8]). Let b , . . . , b s be the Brauercorrespondents of B in C G ( x ) . Let C i be the Cartan matrix of b i for i = 1 , . . . , s . Finally, let C x := C . . . C s ∈ Z l × l . Brauer’s second main theorem yields X t x X x = Y t x Q t x Q x Y x = Y t x C x Y x where X t x denotes the transpose and X x the complex conjugate of X x (see [23, Lemma 5.13]). We mayassume that the entries of X x , Q x , Y x lie in the valuation ring O (recall that these entries are algebraicintegers). It follows from [23, Lemma 2.4 and Theorem 1.19] that Y x is invertible over O . In particular, X t x X x and C x have the same elementary divisors up to multiplication with units in O (recall that O isindeed a principal ideal domain). The largest elementary divisor of C i is the order of a defect group D i of b i and occurs with multiplicity in C i (see [23, Theorem 3.26]). Since D i is conjugate to a subgroupof D , all non-zero elementary divisors of C x are divisors of | D | . Moreover, the number of blocks b i withdefect group D is just the multiplicity of | D | as an elementary divisor of C x .4n the situation of Proposition 5, the pairs ( x, b ) are called ( B -) subsections if b is a Brauer correspondentof B in C G ( x ) . The subsection is called major if b and B have the same defect (group). One mightwonder if the total number of subsections (major or not) can be deduced from the block decompositionof the hermitian matrix X t x X x . However, if B is the only block of G , then X t x X x is just a diagonalmatrix by the second orthogonality relation. Proof of Theorem 2.
The columns of the character table X = X ( G ) corresponding to p -elements aredetermined via [24, Corollary 7.17]. For a p -element x ∈ G , we can decide from X whether there aremajor subsections ( x, b ) by Proposition 5. This happens if and only if x is conjugate to some elementof Z( D ) (see [23, Problem 9.6]). Thus, suppose that x ∈ Z( D ) has order p e and b has defect group D .Then the matrix Q x defined in (2.1) of the previous proof has entries in Q p e . The entries of X x = Q x Y x generate a subfield Q ( X x ) ⊆ Q p e m where m := | G | p ′ . Let b D be a Brauer correspondent of b (and of B ) in D C G ( D ) such that b = b C G ( x ) D . In the following we replace G by the (smaller) Galois group of Q p e m with fixed field Q m . Let γ ∈ G be a non-trivial p -element. By a fusion argument of Burnside, the B -subsections ( x, b ) and ( x γ , b ) are conjugate in G if any only if x and x γ are conjugate in the inertialgroup N G ( D, b D ) (see [23, Problem 9.7]). Since x ∈ Z( D ) and N G ( D, b D ) /D C G ( D ) is a p ′ -group, thiscannot happen. Hence, there exist χ ∈ Irr( B ) and ϕ ∈ IBr( b ) such that d xχ γ ,ϕ = γ ( d xχϕ ) = d x γ χϕ = d xχϕ and χ γ = χ . This shows that Q ( X x ) does not lie in the fixed field of any non-trivial p -element of G .Hence by Galois theory, | Q p e m : Q ( X x ) Q m | is a p ′ -number and | Q ( X x ) Q m : Q m | p = | Q p e m : Q m | p = |G| p = p e − . Therefore, X determines the order of every x ∈ Z( D ) . In particular, exp(Z( D )) is determined.By the proof above, the character table determines whether all x ∈ D are conjugate to elements of Z( D ) . This is a necessary (but insufficient) criterion for D to be abelian. Next we prove the first partof Theorem 3. Proposition 6.
Let B be a p -block of G with abelian defect group D . Let m := | G | p ′ . Then exp( D ) = p | Q ( B ) : Q ( B ) ∩ Q m | p . Proof.
Since D is abelian, all B -subsections are major (see [23, Problem 9.6]). Hence, in the proof ofTheorem 2 there is no need to consider only one p -section at a time. In the end, we can replace Q ( X x ) by Q ( B ) to obtain p | Q ( B ) : Q ( B ) ∩ Q m | p = p | Q ( B ) Q m : Q m | p = exp( D ) . Now we come to the second part of Theorem 3.
Proposition 7.
Let B be a block of G with abelian defect group D and defect at most . Then X ( G ) determines the isomorphism type of D .Proof. Since | D | and exp( D ) are determined by the character table, we may assume that | D | ∈ { p , p } .Let T be the inertial group of some Brauer correspondent of B in C G ( D ) . Since D is abelian, the G -conjugacy classes of B -subsections correspond to the T -orbits on D by [23, Problems 9.6 and 9.7]. Fora fixed x ∈ D , the B -subsections of the form ( x, b ) are pairwise non-conjugate since the blocks areideals of the group algebra of C G ( x ) . Since all B -subsections are major, Proposition 5 allows us to5ount the number of subsections (up to conjugation) corresponding to elements x ∈ D of some fixedorder (note that |h x i| is determined by the proof of Theorem 2). Hence, the character table determinesthe number of T -orbits on D of elements of order p i for each i ≥ .We call x, y ∈ D equivalent if there exist t ∈ T and k ∈ Z such that x t = y kp . Equivalent elementsclearly have the same order. Let d i be the number of equivalence classes of elements in D of order p i for i ≥ . Since T acts coprimely on D , the distinct elements of the form x kp lie in distinct T -orbits.Hence, the number of T -orbits of elements of order p i is d i p i − . In particular, the numbers d i aredetermined by X ( G ) . Note that d is just the number of T -orbits of elements of order p .Now we assume that | D | = p . It suffices to distinguish D ∼ = C p from D ∼ = C p × C p . Suppose first that D ∼ = C p . Then every element of order p in D is a p -power of some element in D . Moreover, if x, y ∈ D are equivalent, so are x p and y p . This shows that d ≤ d . Next we consider D = D × D ∼ = C p × C p .Since T acts coprimely on D , we may assume that D ∼ = C p and D ∼ = C p are T -invariant (see [11,Theorem 5.2.2]). Let x , y ∈ D be of order p . Since D is cyclic, we see that x and y are equivalentif and only if x p and y p are equivalent. For any x , y ∈ D , it follows that x x and y y are equivalentif and only if x p x and y p y are equivalent. Every element of order p has the form x x , but theelements of D do not have the form x p x . Consequently, d > d .It remains to discuss the case | D | = p . If exp( D ) = p , then we need to distinguish C p × C p from C p × C p . But this follows immediately from the case | D | = p above by considering only elementsof order at most p . Hence, we may assume that exp( D ) = p . If D ∼ = C p × C p , we obtain d > d just as in the case C p × C p . Finally, let D = D × D ∼ = C p × C p with T -invariant subgroups D and D . Let ∆ ⊆ D be a T -orbit of elements of order p . Let b ∆ := { y ∈ D : y p ∈ ∆ } . Note that b ∆ is a union of equivalence classes and | b ∆ | = p | ∆ | . Since the size of an equivalence class cannot bedivisible by p , b ∆ contains at least two equivalence classes. For x ∈ ∆ we pick non-equivalent elements b x, e x ∈ b ∆ . Let z = 1 , z , . . . , z s be representatives for the T -orbits in D . Let x, y ∈ D be of order p . Since xz , . . . , xz s lie in distinct T -orbits, we obtain that d − s > s . Moreover, if xz i and yz j arenot equivalent, then b xz i , e xz i , b yz j and e yz j are pairwise non-equivalent elements of order p . Since everyelement of order p outside D is equivalent to some xz i , it follows that d < d − s ) ≤ d .We do not know if our method extends to blocks of defect , but it definitely does not work for defect . In fact, the defect groups C × C and C × C cannot be distinguished by counting orbits of theinertial quotient C ⋊ C (given a suitable action, there are three orbits of involutions and eight orbitsof elements of order in both cases). Nevertheless, these groups can still be distinguished by othermeans.Finally we prove our last theorem. Proof of Theorem 4.
Suppose first that | D/D ′ | = 4 . Then D is a dihedral, a semidihedral or a (gen-eralized) quaternion group by a theorem of Taussky (see [14, Satz III.11.9]). It was shown by Brauerand Olsson that k ( B ) < d (see [37, Theorem 8.1]). They have also computed the generalized decom-position numbers of B , but we only need a small portion of those. For that, let x ∈ D be of order d − and let b x be a Brauer correspondent of B in C G ( x ) . Then b x is a block with cyclic defect group h x i .In particular, b x is nilpotent and IBr( b x ) = { ϕ } . If D is a dihedral or a quaternion group, then x isconjugate to x − in D . From the structure of the fusion system of B (see [37, Theorem 8.1]) we see thatthere is no more fusion inside h x i . It follows as in Theorem 2 that Q ( d xχϕ : χ ∈ Irr( B )) = Q ( ζ + ζ − ) .In the semidihedral case we obtain similarly that Q ( d xχϕ : χ ∈ Irr( B )) = Q ( ζ − ζ − ) .Next let y ∈ D be arbitrary. If y has order at most , then the generalized decomposition numberswith respect to y are rational integers. Thus, we may assume that |h y i| > . Then y and y − (or6 − d − if D is semidihedral) are conjugate in D and the Brauer correspondent b y has cyclic defectgroup (namely h y i or h x i ; see [37, Lemma 1.34]). Hence, IBr( b y ) = { µ } and the argument above yields d yχµ ∈ Q ( ζ ± ζ − ) for every χ ∈ Irr( B ) . Therefore, the entries of the generalized decomposition matrix Q of B generate the field Q ( Q ) = Q ( ζ ± ζ − ) ⊆ Q d . Let m := | G | ′ and γ ∈ Gal( Q d | Q ) . Let b γ be theunique extension of γ to Gal( Q d m | Q m ) . Then γ ∈ Gal( Q d | Q ( Q )) ⇐⇒ b γ ∈ Gal( Q d m | Q ( Q ) Q m ) = Gal( Q d m | Q ( B ) Q m ) ⇐⇒ γ ∈ Gal( Q d | Q ( B ) Q m ∩ Q d ) (this argument is due to Reynolds [35]). The main theorem of Galois theory implies Q ( B ) Q m ∩ Q d = Q ( Q ) = Q ( ζ ± ζ − ) as desired.Now assume conversely that k ( B ) < d and Q ( B ) Q m ∩ Q d = Q ( ζ ± ζ − ) . If D is cyclic or of type C d − × C with d ≥ , then B is nilpotent in contradiction to k ( B ) < d . Suppose that exp( D ) < d − .Then the generalized decomposition numbers of B lie in Q d − and we obtain ζ ± ζ − ∈ Q ( B ) Q m ∩ Q d ⊆ Q d − m ∩ Q d = Q d − . This forces d = 3 and exp( D ) = 2 . Then however, D is elementary abelian and k ( B ) = 8 = 2 d byKessar–Koshitani–Linckelmann [15]. This contradiction shows that exp( D ) = 2 d − . Now it is well-known that | D : D ′ | = 4 unless d > and D = h x, y : x d − = y = 1 , yxy − = x d − i . In this exception, B is nilpotent by [37, Theorem 8.1]. By Broué–Puig [4], there exists a -rationalcharacter χ ∈ Irr( B ) such that Irr( B ) = { λ ∗ χ : λ ∈ Irr( D ) } . This yields the contradiction Q ( B ) Q m = Q ( D ) Q m = Q d − m . For the last claim recall that B has tame representation type if and only if D is a Klein four-group(detectable by Theorem 3) or D is non-abelian and | D/D ′ | = 4 .We remark that the distinction of the defect groups of order in the proof above relies implicitly on theclassification of finite simple groups (via [15]). The dependence on the CFSG can be avoided by makinguse of the remark after the proof of Theorem 2. As in [25], the Alperin–McKay Conjecture would implythat | D/D ′ | = 4 if and only if B has exactly four irreducible characters of height (provided p = 2 ). As before, let B be a p -block of G with defect group D . Let X ( B ) be the submatrix of X ( G ) withrows indexed by Irr( B ) . By the block orthogonality relation (see [23, Corollary 5.11]), the matrix X ( B ) t X ( B ) has block diagonal shape. The blocks (of that matrix) are the matrices X t x X x studied inthe proof of Proposition 5. Let (1 , B ) = ( x , b ) , . . . , ( x s , b s ) be representatives for the G -conjugacyclasses of B -subsections. Let C i be the Cartan matrix of b i . We have seen in the proof of Proposition 5that X ( B ) t X ( B ) and the block diagonal matrix C ⊕ . . . ⊕ C s have the same non-zero elementarydivisors e , . . . , e k over O (up to multiplication with units in O ). Hence, we may assume that e , . . . , e k are uniquely determined integer p -powers. We call e , . . . , e k the elementary divisors of B . It turns outthat these numbers are the orders of the lower defect groups of B (with multiplicities) introduced byBrauer [2] (see Proposition 11 below). We call γ ( B ) := 1 e + . . . + 1 e k ∈ Q fusion number of B . This definition is inspired by the class equation in finite groups as will becomeclear in the sequel. Conjecture 8.
For every block B of G we have γ ( B ) ≥ with equality if and only if B is nilpotent. In contrast to the character degrees considered in [22], we will see that the fusion number is invariantunder categorical equivalences like isotypies.To verify that γ ( B ) ≥ , it is often enough to consider only the Cartan matrix C of B . If C possessesan entry coprime to p , then is an elementary divisor and γ ( B ) ≥ with equality if and only if B has defect . The remaining elementary divisors e i > can in principle be computed locally (see [32,Theorem 4.3]).Before providing evidence for Conjecture 8, we derive a consequence which strengthens Puig’s Conjec-ture (mentioned in the introduction) and was established for abelian defect groups in [34]. Proposition 9.
Conjecture 8 implies that B is nilpotent if and only if l ( b ) = 1 for every B -subsection ( x, b ) .Proof. It is well-known that every nilpotent block fulfills the condition. Suppose conversely that l ( b ) = 1 for every B -subsection ( x, b ) . Let D be a defect group of B and let b D be a Brauer correspondent of B in D C G ( D ) . By [37, Lemma 1.34], there exist representatives ( x , b ) , . . . , ( x s , b s ) ∈ ( D, b D ) for the G -conjugacy classes of B -subsections such that b i is uniquely determined by x i and has defect group C D ( x i ) . Since l ( b i ) = 1 , the Cartan matrix of b i is ( | C D ( x i ) | ) for i = 1 , . . . , s . The elements x , . . . , x s can be complemented to a set of representatives x , . . . , x t for the conjugacy classes of D . The classequation for D shows that γ ( B ) = s X i =1 | C D ( x i ) | ≤ t X i =1 | C D ( x i ) | = 1 . According to Conjecture 8, γ ( B ) = 1 and B is nilpotent. Theorem 10.
Conjecture 8 holds in each of the following situations:(i) B is nilpotent.(ii) B is the only block of G .(iii) B has cyclic defect group.(iv) G is a symmetric group.(v) G is a simple group of Lie type in defining characteristic.(vi) G is a quasisimple group appearing in the ATLAS.Proof. (i) This follows from the proof of Proposition 9. For the remaining parts we may assume that B isnon-nilpotent. 8ii) If B is the only block of G , then X ( B ) = X ( G ) = X and X t X = diag( | C G ( g i ) | : i = 1 , . . . , k ) where g , . . . , g k represent the conjugacy classes of G . In particular, e i = | C G ( g i ) | p for i = 1 , . . . , k .The class equation of G reads | C G ( g ) | + . . . + 1 | C G ( g k ) | = 1 . Hence, γ ( B ) > unless G is a p -group in which case B is nilpotent.(iii) It is well-known that a non-nilpotent block with cyclic defect group has elementary divisor (see[37, Theorem 8.6]). This implies γ ( B ) > as explained before Proposition 9.(iv) Let B be a p -block of weight w of the symmetric group S n . It is well-known that B is nilpotentif and only if w = 0 or ( p, w ) = (2 , . Thus, let w ≥ . If p is odd or w is even, then C haselementary divisor by a theorem of Olsson [33, Corollary 3.13]. Now let p = 2 and w = 2 k +1 ≥ .Then the multiplicity of as an elementary divisor of C is the number of partitions of w withexactly one odd part (this can be extracted from [1, Theorem 4.5]). Since there are at least twosuch partitions (namely ( w ) and (2 k, ), it follows that γ ( B ) > .(v) Apart from finitely many exceptions (like F (2) ′ ) which are covered by (vi) below, we mayassume that G has only two blocks: the principal block B and a block of defect containingthe Steinberg character (see [13, Section 8.5]). Malle [21, Corollary 4.2] has shown that there areat least two non-conjugate elements g, h ∈ G such that | C G ( g ) | p = | C G ( h ) | p = 1 . One of themaccounts for an elementary divisor of the Cartan matrix of B . Thus, γ ( B ) > .(vi) In order to check the claim by computer, we replace B by the union of its Galois conjugate blocksso that X ( B ) t X ( B ) becomes an integral matrix. The p -parts of the elementary divisors of thatmatrix can be computed efficiently with Frank Lübeck’s edim package [19] for GAP [10]. SinceGalois conjugate blocks clearly have the same fusion number, we need to divide by the number ofGalois conjugate blocks in the end. It turns out that for all blocks B of quasisimple groups in theATLAS, γ ( B ) > unless all characters have the same degree. In the latter case, B is nilpotentby [31] (in fact, An and Eaton have shown that all nilpotent blocks of quasisimple groups haveabelian defect groups).We have also compared Conjecture 8 to the Malle–Navarro Conjecture (mentioned in the introduction)for small groups ( | G | ≤ ) without finding any differences.In the remainder of the paper we offer two reduction theorems. To this end, we review Olsson’s work [32]on lower defect groups which makes use of the algebraically closed field F := O / J( O ) of characteristic p . We denote the set of blocks of G by Bl( G ) and the set of conjugacy classes by Cl( G ) . For B ∈ Bl( G ) let ǫ B be the block idempotent of B as a subalgebra of F G . Moreover, K + := P x ∈ K x ⊆ Z( F G ) is theclass sum of K ∈ Cl( G ) . Proposition 11.
The set
Cl( G ) can be partitioned into a so-called block splitting Cl( G ) = [ B ∈ Bl( G ) Cl( B ) such that { K + ǫ B : K ∈ Cl( B ) } is a basis of Z( B ) ⊆ Z( F G ) for every B ∈ Bl( G ) . If x K ∈ K ∈ Cl( B ) , then the Sylow p -subgroups of C G ( x K ) are called lower defect groups of B . Their orders arethe elementary divisors of B , i. e. { e , . . . , e k } = {| C G ( x K ) | p : K ∈ Cl( B ) } as multisets. roof. The existence of block splittings is proved in [32, Proposition 2.2] (the proof is revisited inthe following lemma). To verify the second claim, we freely use the notation from [32]. In particular, m B ( P ) denotes the multiplicity of a p -subgroup P ∈ P ( G ) as a lower defect group of B . By combiningTheorems 3.2, 5.4(1) and Corollary 7.7 of [32], the multiplicity of p n in the multiset {| C G ( x K ) | p : K ∈ Cl( B ) } is X P ∈P ( G ) | P | = p n m B ( P ) = X P ∈P ( G ) | P | = p n X x ∈ Π( G ) m ( x ) B ( P ) = X x ∈ Π( G ) X b ∈ Bl(C G ( x )) b G = B X Q ∈P (C G ( x )) | Q | = p n m (1) b ( Q ) . Now by [32, Remark on p. 285], X Q ∈P (C G ( x )) | Q | = p n m (1) b ( Q ) is the multiplicity of p n as an elementary divisor of the Cartan matrix of b . Moreover, every B -subsection ( x, b ) appears (up to G -conjugation) just once in the sum.In the language of block splittings our conjecture can be rephrased as X K ∈ Cl( B ) | K | p ≥ | G | p with equality if and only if B is nilpotent. Lemma 12.
Let Z ≤ Z( G ) be of order p . Then there exists a block splitting Cl( G ) = S B ∈ Bl( G ) Cl( B ) such that Cl( B ) = { Kz : K ∈ Cl( B ) } for all B ∈ Bl( G ) and z ∈ Z .Proof. In order to exploit Olsson’s proof of the existence of block splittings, we recall the full details.Instead of the generalized Laplace expansion we make use of the Leibniz formula for determinants. Let
Bl( G ) = { B , . . . , B n } and Cl( G ) = { K , . . . , K k } . Let I ∪ . . . ∪ I n = { , . . . , k } be a partition suchthat { b i : i ∈ I j } is an F -basis of Z( B j ) . Then b , . . . , b k is a basis of Z( B ) ⊕ . . . ⊕ Z( B n ) = Z( F G ) .On the other hand, the class sums K +1 , . . . , K + k also form a basis of Z( F G ) . Hence, there exists aninvertible matrix A = ( a ij ) ∈ F k × k such that A b ... b k = K +1 ... K + k . (3.1)Let J = ( J , . . . , J n ) be a partition of { , . . . , k } such that | J i | = | I i | for i = 1 , . . . , n . Let σ J ∈ S n bethe unique permutation which sends J i to I i for i = 1 , . . . , n and preserves the natural order of thosesets. By the Leibniz formula, = det A = X α ∈ S k sgn( α ) k Y i =1 a i,α ( i ) = X J X α ∈ S I × ... × S In sgn( ασ J ) k Y i =1 a i,ασ J ( i ) = X J sgn( σ J ) (cid:16) X α ∈ S I sgn( α ) Y j ∈ J a j,α σ J ( j ) (cid:17) . . . (cid:16) X α n ∈ S In sgn( α n ) Y j ∈ J n a j,α n σ J ( j ) (cid:17) = X J sgn( σ J ) det( A J I ) . . . det( A J n I n ) (3.2)10here A I i J i := ( a st : s ∈ I i , t ∈ J i ) . Hence, there exists some partition J such that det( A J i I i ) = 0 for i = 1 , . . . , n . We now multiply (3.1) with the block idempotent ǫ s of B s to get A J s I s ( b i : i ∈ I s ) = ( K + j ǫ s : j ∈ J s ) (notice that b j ǫ s = 0 if j / ∈ I s ). Hence, the sets Cl( B s ) := { K j : j ∈ J s } form a block splitting of G .Next we observe that Z = h z i acts by multiplication on Cl( G ) . In this way, z induces a permutation π z on { , . . . , k } such that K j z = K π z ( j ) . Let J ′ i := π z ( J i ) for i = 1 , . . . , n . We claim that J ′ makes thesame contribution to (3.2) as J . Since Z( B s ) is an ideal of Z( F G ) , we see that ( K + j zǫ s : j ∈ J s ) is abasis of Z( B s ) z = Z( B s ) . Thus, there exists A s ∈ GL( k ( B s ) , F ) such that A s ( K + j ǫ s : j ∈ J s ) = ( K + j zǫ s : j ∈ J s ) . Since z p = 1 , it follows that A ps = 1 . In particular det( A s ) = 1 , since F has characteristic p . Let τ s ∈ S J ′ s such that elements τ s π z ( j ) with j ∈ J s appear in their natural order. Let P s be the permutationmatrix corresponding to τ s . Then A J ′ s I s = P s A s A J s I s for s = 1 , . . . , n . In particular, det( A J ′ s I s ) =sgn( τ s ) det( A J s I s ) for s = 1 , . . . , n . If p = 2 , it is now clear that J and J ′ make the same contributionto (3.2). Thus, let p > . Then π z has order p and therefore sgn( π z ) = 1 . Moreover, σ J ′ τ . . . τ s π z = σ J .Consequently, sgn( σ J ′ ) det( A J ′ I ) . . . det( A J ′ n I n ) = sgn( σ J ) det( A J I ) . . . det( A J n I n ) as desired.If J ′ = J , then the orbit of J under Z has length p . The corresponding p equal summands of (3.2)cancel out. Since we still have det A = 0 , there must exist a block splitting J such that J ′ = J . Theclaim follows.In the remark after [32, Propsosition 7.8] Olsson states that there is no relation between a lower defectgroup of a block and its dominated block modulo a central p -subgroup. Nevertheless, we show thatthere is a relation if one considers all lower defect groups at the same time. Proposition 13.
Let Z be a p -subgroup of Z( G ) . Let B be a p -block of G and let B be the unique blockof G/Z dominated by B . Then γ ( B ) = γ ( B ) and B is nilpotent if and only if B is.Proof. The second claim was proved in [40, Lemma 2]. A modern proof in terms of fusion systemscan be given along the following lines. The fusion system F of B contains Z in its center, i. e. F =C F ( Z ) . One then shows that F := F /Z is the fusion system of B (see [7, Definition 5.9]). Now by [7,Proposition 5.60], there is a one-to-one correspondence between F -essential subgroups and F -essentialsubgroups. Hence, B is nilpotent if and only if B is.To prove the first claim, we may assume that | Z | = p by induction on | Z | . It is convenient to provethe claim for all blocks B ∈ Bl( G ) at the same time. By Lemma 12, there exists a block splitting Cl( G ) = S B ∈ Bl( G ) Cl( B ) such that K ∈ Cl( B ) ⇐⇒ Kz ∈ Cl( B ) (3.3)for all K ∈ Cl( G ) and z ∈ Z .The canonical epimorphism G → G := G/Z maps ǫ B to ǫ B = ǫ B since B is the only block dominatedby B (see [23, p. 198]). Moreover, K ∈ Cl( G ) for every K ∈ Cl( G ) . Hence, { K + ǫ B : K ∈ Cl( B ) } spans Z( B ) . If K, L ∈ Cl( G ) induce the same class K = L , then L = Kz for some z ∈ Z . In this11ase (3.3) implies that K ∈ Cl( B ) ⇐⇒ L ∈ Cl( B ) . Thus, after removing duplicates from the set Cl( B ) := { K : K ∈ Cl( B ) } , we obtain a partition Cl( G ) = [ B ∈ Bl( G ) Cl( B ) . Since | Cl( G ) | = dim Z( F G ) = X B ∈ Bl( G ) dim Z( B ) ≤ X B ∈ Bl( G ) | Cl( B ) | = | Cl( G ) | , the sets Cl( B ) form a block splitting of Cl( G ) .Finally, we determine the elementary divisors e , . . . , e k of B . By Proposition 11, we may label Cl( B ) = { K , . . . , K k } such that e i | K i | p = | G | p for i = 1 , . . . , k . If | K i | = | K i | , then the p classes K i z ∈ Cl( B ) with z ∈ Z are all distinct. Since p e i | K i | p = | G | p , we have p of the e i , say e i = . . . = e i p accountingfor one elementary divisor e i := p e i of B . If, on the other hand, | K i | = p | K i | , then K i = K i z for all z ∈ Z . In this case we set e i := e i . This gives the elementary divisors e , . . . , e l of B such that γ ( B ) = 1 e + . . . + 1 e l = 1 e + . . . + 1 e k = γ ( B ) . Our final result is a reduction for blocks of p -solvable groups to a purely group-theoretic assertion (itmight be called a projective class equation). Proposition 14.
Conjecture 8 holds for all p -blocks of p -solvable groups if and only if the following istrue: Let G be a p -solvable group such that Z := Z( G ) = O p ′ ( G ) ≤ G ′ is cyclic and Z = O p ( G ) . Let K , . . . , K n be the conjugacy classes of G/Z consisting of elements xZ such that C G/Z ( xZ ) = C G ( x ) /Z .Then | K | p + . . . + | K n | p > | G | p .Proof. Let B be a p -block of a p -solvable group G . By Broué [3, Théorème 5.5], B is isotypic to ablock of a p -solvable group H such that O p ′ ( H ) ⊆ Z( H ) . Since isotypies preserve the generalizeddecomposition matrices up to basic sets (see [3, Théorème 4.8]), also the elementary divisors of B are preserved. Hence, we may assume that Z := O p ′ ( G ) ≤ Z( G ) . By Proposition 13, we may furtherassume that Z = Z( G ) . Recall that Ker( B ) ≤ Z by [23, Theorem 6.10]. Obviously, B is nilpotent ifand only if the isomorphic block B of G/ Ker( B ) is nilpotent. Moreover, X ( B ) is obtained from X ( B ) by removing duplicate columns. It follows that γ ( B ) = γ ( B ) . By replacing G with G/ Ker( B ) , wemay assume that B is faithful. By Theorem 10, we may assume that B is non-nilpotent and therefore Z = O p ( G ) . The reduction to Z ≤ G ′ will be established at the end of the proof.In order to construct a block splitting, we need to consider all blocks of G . By [23, Theorem 10.20], theblocks of G can be labeled by λ ∈ Irr( Z ) such that Irr( B λ ) = Irr( G | λ ) . The block idempotent of B λ isjust the ordinary character idempotent ǫ λ ∈ Z( F Z ) (see [23, p. 51]). Note that Z acts by multiplicationon Cl( G ) . Let Z K := { z ∈ Z : zK = K } ≤ Z be the stabilizer of K ∈ Cl( G ) . The classes in the orbitof K can be labeled arbitrarily by Irr(
Z/Z K ) ≤ Irr( Z ) , say { Kz : z ∈ Z } = { K λ : λ ∈ Irr(
Z/Z K ) } .We define Cl( B λ ) := { K λ : K ∈ Cl( G ) , Z K ⊆ Ker( λ ) } . Note that for K ∈ Cl( B λ ) and z ∈ Z we have ( Kz ) + ǫ λ = λ ( z ) K + ǫ λ ∈ F · K + ǫ λ . On the other hand, if z ∈ Z K \ Ker( λ ) , then K + ǫ λ = ( Kz ) + ǫ λ = λ ( z ) K + ǫ λ = 0 .
12t follows easily that Z( B λ ) is spanned by { K + ǫ λ : K ∈ Cl( B λ ) } . Since | Cl( G ) | = dim Z( F G ) = X λ ∈ Irr( Z ) dim Z( B λ ) ≤ X λ ∈ Irr( Z ) | Cl( B λ ) | = | Cl( G ) | , we conclude that Cl( G ) = S λ ∈ Irr( Z ) Cl( B λ ) is indeed a block splitting of G (this can also be explainedwith the notion of good conjugacy classes in [24, Theorem 5.14]).We only need to verify the claim for a faithful block B = B λ , i. e. Ker( λ ) = 1 and Z is cyclic. Herethe conjugacy classes K ∈ Cl( B ) represent the regular orbits of Z on Cl( G ) . Thus, for x ∈ K wehave C G ( x ) /Z = C G/Z ( xZ ) as desired. Now we fix coset representatives b g for every g ∈ G/Z . Thenthe equation b g b h = α ( g, h ) c gh where g, h ∈ G/Z defines a 2-cocycle α ∈ Z ( G/Z, Z ) . Let β := λ ◦ α ∈ Z ( G/Z, F × ) . It is well-known that the map g b gǫ λ induces an algebra isomorphism between thetwisted group algebra F β [ G/Z ] and B . The class sums K + ǫ with K ∈ Cl( B ) correspond to the so-called β - regular class sums of F β [ G/Z ] (these are the only non-vanishing class sums in F β [ G/Z ] andtherefore form a basis of Z( F β [ G/Z ]) ).Since β can be regarded as an element of the Schur multiplier H ( G/Z, F × ) , F β [ G/Z ] is also isomorphicto a faithful block of a covering group e G with cyclic e Z ≤ Z( e G ) ∩ e G ′ such that e G/ e Z ∼ = G/Z . Againthe β -regular class sums correspond to the regular orbits of e Z on Cl( e G ) . Moreover, we still have e Z = O p ′ ( e G ) = O p ( e G ) . Hence, we may replace G by e G and Z by e Z . Since B is non-nilpotent, it remainsto show that X K ∈ Cl( B ) | K | p = | G | p γ ( B ) > | G | p . A concrete example to Proposition 14 is the double cover of S × S for p = 3 . Here the fusion numberof the unique non-principal block is / (this is the smallest number larger than that we haveencountered).If Conjecture 8 can be verified for blocks of p -solvable groups, then it also holds for blocks withnormal defect groups since such blocks are splendid Morita equivalent to blocks of p -solvable groupsby Külshammer [18]. Similarly, Conjecture 8 would follow for blocks with abelian defect groups ifadditionally Broué’s Conjecture is true. Acknowledgment
I thank Gabriel Navarro for stimulating discussions on this paper, Christine Bessenrodt for makingme aware of [1] and Gunter Malle for providing [21]. Moreover, I appreciate a very careful reading ofan anonymous referee. The work is supported by the German Research Foundation (SA 2864/1-2 andSA 2864/3-1).
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