Morita equivalence classes of blocks with elementary abelian defect groups of order 16
aa r X i v : . [ m a t h . G R ] M a y Morita equivalence classes of blocks with elementaryabelian defect groups of order 16 ∗ Charles W. Eaton † Abstract
We classify the Morita equivalence classes of blocks with elementary abeliandefect groups of order 16 with respect to a complete discrete valuation ring withalgebraically closed residue field of characteristic two. As a consequence, blockswith this defect group are derived equivalent to their Brauer correspondent inthe normalizer of a defect group and so satisfy Brou´e’s Conjecture.Keywords: Donovan’s conjecture; Morita equivalence; finite groups; blocktheory.
Throughout let k be an algebraically closed field of prime characteristic ℓ and let O bea discrete valuation ring with residue field k and field of fractions K of characteristiczero. We assume that K is large enough for the groups under consideration. Weconsider blocks B of O G with defect group D , for finite groups G .Our purpose is the description of the Morita and derived equivalence classes of(module categories for) blocks of finite groups with a given defect group. It is alreadyknown by [13] that Donovan’s conjecture holds for elementary abelian 2-groups, thatis, for each n ∈ N there are only finitely many Morita equivalence classes of blockswith defect group ( C ) n , and so in theory Morita equivalence classes of such blockscould be classified for any given n . Here we consider the case n = 4 and achievea complete classification. The main tool is the description given in [13] of the 2-blocks with abelian defect groups of the quasisimple groups. The number of irreducibleordinary and Brauer characters of blocks with defect group ( C ) has already beendetermined in [29] and [13]. Our work continues [12] in which a classification is givenfor blocks with elementary abelian defect groups of order 8. The Morita equivalenceclasses of block with Klein four defect groups are known by [18] and [32]. Other ℓ -groupswhere there are classifications are: cyclic ℓ -groups, where the Morita equivalence classescan be characterised in terms of Brauer trees (in work by many, for which see [33]); ∗ This research was supported by the EPSRC (grant no. EP/M015548/1). † School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom. Email:[email protected] k ); nonmetacylic minimal nonabelian 2-groups h x, y : x r = y s = [ x, y ] = [ x, [ x, y ]] = [ y, [ x, y ]] = 1 i , where r ≥ s ≥
1, by [40]and [14]; 2-groups which are a direct product of cyclic factors all of different orders,whose automorphism groups are themselves 2-groups so the block must be nilpotentand so Morita equivalent to O D by [6] and [37]; the remaining metacyclic 2-groupsnot listed above, which force the block to be nilpotent by [8]. Finally, principal blockswith defect group C × C are classified (even up to Puig equivalence) in [23].A significant challenge arises in our situation which does not arise for defect groupsof order 8 in that we must address Morita equivalences in the case of a normal subgroupof index 3 where there are infinitely many possibilities for N but the Morita equivalenceclass of the block of N is fixed. We do so by following K¨ulshammer’s analysis in [27]of possible crossed products in detail.Before stating the main theorem, we recall the definition of a subpair and of theinertial quotient of B .A B -subpair is a pair ( Q, b Q ) where Q is a p -subgroup of G and b Q is a block of O QC G ( Q ) with Brauer correspondent B . When D is a defect group for B , the B -subpairs ( D, b D ) are G -conjugate. Write N G ( D, b D ) for the stabilizer in N G ( D ) of b D .Then the inertial quotient of B is E = N G ( D, b D ) /DC G ( D ), an ℓ ′ -group unique upto isomorphism. To simplify further definitions, suppose that D is abelian. We say( Q, b Q ) ≤ ( R, b R ) for subpairs ( Q, b Q ) and ( R, b R ) if Q ≤ R and ( b R ) C G ( Q ) = b Q . If x ∈ D and b x is a block of C G ( x ) we say that the Brauer element ( x, b x ) ∈ ( D, b D ) if( b D ) C G ( x ) = b x .Note that if D is abelian, then B is nilpotent precisely when the inertial quotientis trivial.The possible inertial quotients for a block with defect group D are given in [29],and these are: 1 (corresponding to nilpotent blocks); C with action as in A × ( C ) ; C consisting of 5th powers of a Singer cycle for F (with only one fixed point in itsaction on D ); C consisting of 3rd powers of a Singer cycle; C coming from a Singercycle for F ; C × C ; C ⋊ C coming from a Singer cycle and a field automorphism of F ; C coming from a Singer cycle of F . Each inertial quotient apart from C × C has trivial Schur multiplier, and in this case it is C .We say that a block with defect group ( C ) is of type E if it has inertial quotient E where there is only one possible faithful action on ( C ) , and in the case that theinertial quotient is C , we say it has type ( C ) when the action is as in A × ( C ) andtype ( C ) when there is only one fixed point. Theorem 1.1
Let B be a block of O G with elementary abelian defect group D oforder , where G is a finite group. Then B is Morita equivalent to precisely one ofthe following:(a) a non-principal block of ( C ) ⋊ , where the centre of acts trivially, andwe note that the two non-principal blocks are Morita equivalent;(b) the principal block of precisely one of the following:(i) D ; ii) ( C ) × A ;(iii) ( C ) × A ;(iv) D ⋊ C of type ( C ) ;(v) D ⋊ C ;(vi) C × (( C ) ⋊ C ) ;(vii) C × SL (8) (type C );(viii) A × A ;(ix) A × A (type C × C );(x) A × A (type C × C );(xi) D ⋊ C ;(xii) SL (16) (type C );(xiii) C × (( C ) ⋊ ( C ⋊ C )) ;(xiv) C × J (type C ⋊ C );(xv) C × Aut( SL (8)) (type C ⋊ C ).If B is a principal block, then it is Morita equivalent to one of the examples in case(b), i.e., the blocks in case (a) cannot be Morita equivalent to a principal block of anyfinite group.Blocks are derived equivalent if and only if they have the same inertial quotient andnumber of simple modules. Remarks 1.2 (i) It will be clear from the proof of Theorem 1.1 that blocks with defectgroup ( C ) cannot be Morita equivalent to a block with non-isomorphic defect group.The same can be said of the inertial quotient, as the number of irreducible charactersand number of simple modules together determine the inertial quotient.(ii) Non-nilpotent blocks with defect group ( C ) and just one simple module arestudied in [31]. The structure of the centre of such a block is described there, andhopefully in the future there will be a classification-free proof that all such blocks areMorita equivalent to the blocks in part (a) of Theorem 1.1. Corollary 1.3
Broue’s abelian defect group conjecture holds for blocks with defectgroup ( C ) , that is, if B is a block of O G with such a defect group D , then B isderived equivalent to its Brauer correspondent in O N G ( D ) . Remark 1.4
We cannot say at present whether the blocks are splendid equivalent as wecannot say anything about the sources of the bimodules giving the Morita equivalencesin Theorem 1.1.
The paper is structured as follows. In Section 2 we give many of the miscellaneouspreliminary results necessary for the proof of Theorem 1.1. In Section 3 we give thenecessary background on crossed products and Picard groups, and analyse possiblecrossed products in situations that will arise in the proof of the main theorem. InSection 4 we prove Theorem 1.1. 3
Preliminary results
Let G be a finite group, N ✁ G and let b be a G -stable block of O N . The normalsubgroup G [ b ] of G is defined to be the group of elements of G acting as inner auto-morphisms on b ⊗ O k . We first collect some results concerning G [ b ] that will be usedwhen considering automorphism groups of simple groups. Proposition 2.1
Let G be a finite group and B a block of O G with defect group D .Let N ✁ G with D ≤ N and suppose that B covers a G -stable block b of O N . Let B ′ be a block of O G [ b ] covered by B . Then(i) b is source algebra equivalent to B ′ , and in particular has isomorphic inertialquotient;(ii) B is the unique block of O G covering B ′ . Proof . Part (i) is [22, 2.2], noting that a source algebra equivalence over k impliesone over O by [37, 7.8]. Part (ii) follows from [11, 3.5]. ✷ The following is a distillation of those results in [28] which are relevant here.
Proposition 2.2 ([28])
Let G be a finite group and N ✁ G . Let B be a block of O G with defect group D covering a G -stable nilpotent block b of O N with defect group D ∩ N . Then there is a finite group L and M ✁ L such that (i) M ∼ = D ∩ N , (ii) L/M ∼ = G/N , (iii) there is a subgroup D L of L with D L ∼ = D and D L ∩ M ∼ = D ∩ N ,and (iv) there is a central extension ˜ L of L by an ℓ ′ -group, and a block ˜ B of O ˜ L whichis Morita equivalent to B and has defect group ˜ D ∼ = D L ∼ = D .If B is the principal block, then ˜ B is the principal block. Proof . Guidance on the extraction of these results form [28] is given in [12, 2.2].It remains to prove the claim regarding the principal block. Note that if B is theprincipal block, then b is also principal and so N has a normal ℓ -complement. Then O ℓ ′ ( N ) lies in the kernel of B and the corresponding ˜ B is the principal block. ✷ Recall that a block of a finite group G is quasiprimitive if every block of everynormal subgroup that it covers is G -stable under conjugation. Corollary 2.3
Let G be a finite group and N ✁ G with N Z ( G ) O ℓ ( G ) . Let B bea quasiprimitive block of O G with defect group D covering a nilpotent block b of O N .Then there is a finite group H with [ H : O ℓ ′ ( Z ( H ))] < [ G : O ℓ ′ ( Z ( G ))] and a block B H with defect group D H ∼ = D such that B H is Morita equivalent to B . Proof . Let b ′ be the block of O Z ( G ) N covered by B and covering b . Then b ′ mustalso be nilpotent, and we may assume that Z ( G ) ≤ N . Applying Proposition 2.2, wemay take H = ˜ L and B H = ˜ B . Note that [ ˜ L : O ℓ ′ ( Z ( ˜ L ))] ≤ | L | = [ G : N ] | D ∩ N | < [ G : O ℓ ′ ( Z ( G ))]. ✷ Lemma 2.4
Let G be a finite group and let N ✁ G with G/N ℓ -solvable. Let B be aquasiprimitive block of G with abelian defect group D . Then DN/N ∈ Syl ℓ ( G/N ) . roof . There is a series N = G ✁ G ✁ · · · ✁ G n = G with each G i normal in G and each quotient either an ℓ -group or an ℓ ′ -group. Let B i be the unique block of O G i covered by B . If G i +1 /G i is an ℓ ′ -group, then B i +1 and B i share a defect group. Supposethat G i +1 /G i is an ℓ -group. Then B i +1 is the unique block of O G i +1 covering B i , andso by [1, Theorem 15.1] a defect group D i +1 of B i +1 satisfies D i +1 G i /G i = G i +1 /G i .The result follows. ✷ Proposition 2.5 ([42])
Let B be an ℓ -block of O G for a finite group G and let Z ≤ O ℓ ( Z ( G )) . Let ¯ B be the unique block of O ( G/Z ) corresponding to B . Then B isnilpotent if and only if ¯ B is nilpotent. Proof . The result in [42] is stated over k , but it follows over O immediately. ✷ Recall that a block B of O G is nilpotent covered if there is a finite group H with G ✁ H and a nilpotent block of O H covering B . Let D be a defect group for B andlet b be the Brauer correspondent of B in O N G ( D ). Following [38] B is inertial if itis basic Morita equivalent to b , that is, if there is a Morita equivalence induced by abimodule with endopermutation source. Proposition 2.6 ([38], [46])
Let G and N be finite groups and N ✁ G . Let b be ablock of O N covered by a block B of O G .(i) If B is inertial, then b is inertial.(ii) If b is nilpotent covered, then b is inertial.(iii) If ℓ [ G : N ] and b is inertial, then B is inertial.(iv) If b is nilpotent covered, then it has abelian inertial quotient. Proof . (i) is [38, Theorem 3.13], (ii) and (iv) are [38, Corollary 4.3]. (iii) is themain theorem of [46]. ✷ We will make frequent use of the classification of Morita equivalence classes ofblocks with Klein four defect groups throughout this paper without further reference:
Proposition 2.7 ([18], [32], [7])
Let B be a block of O G for a finite group G . If B has Klein four defect group D , then it is source algebra equivalent to the principal blockof one of O D , O A and O A . We extract the results of [13] necessary for this paper:
Proposition 2.8 ([13])
Let B be a block of O G for a quasisimple group G with ele-mentary abelian defect group D of order dividing . Then one or more of the followingoccurs:(i) G ∼ = SL (16) , J or G ( q ) , where q = 3 m +1 for some m ∈ N , and B is theprincipal block;(ii) G ∼ = Co and B is the unique non-principal -block of defect ;(ii) G is of type D n ( q ) or E ( q ) for some q of odd prime power order, O ( G ) = 1 and B is Morita equivalent to a block C of a O L where L = L × L ≤ G such that L is abelian and the block of O L covered by C has Klein four defect groups;(iii) | O ( G ) | = 4 and D/O ( G ) is a Klein four group;(iv) B is nilpotent covered. roof . This follows from Proposition 5.3 and Theorem 6.1 of [13]. ✷ One obstacle in classifying Morita equivalence classes over O rather than k is thatthe results of [24] only apply over k . However in our situation we are lucky to be ableto apply some work of Watanabe on perfect isometries to obtain the same result over O in certain crucial cases. For the benefit of the reader we state the relevant resultof [43] here. First we need some more notation.Let B be a block of O G , where G is a finite group. Write L K ( G, B ) for the group ofgeneralized characters of B with respect to K . Let χ be a generalized character of B .Fix a maximal B -subpair ( D, b D ). Let λ be a generalized character of a defect group D of B such that whenever ( x, b x ) ∈ ( D, b D ) and z ∈ G such that ( x, b x ) z ∈ ( D, b D ),we have λ ( x ) = λ ( x z ). Define λ ∗ χ as in [5], another generalized character of B . In thefollowing, if λ is a generalized character of a factor group of D , then we are implicitlyconsidering its inflation to D . Proposition 2.9 (Lemma 3 of [43])
Let B be a block of a finite group G coveringa G -stable block b of N ✁ G . Suppose that B has an abelian defect group D and thereis Q ≤ D such that D = Q × ( D ∩ N ) and G = N ⋊ Q . Let b D be a block of C G ( D ) with Brauer correspondent B , and write B ′ = ( b D ) C G ( Q ) . If there is a perfect isometry I : L K ( C G ( Q ) , B ′ ) → L K ( G, B ) satisfying I ( λ ∗ ζ ) = λ ∗ I ( ζ ) for all λ ∈ Irr( Q ) and ζ ∈ Irr( B ′ ) , then B ∼ = O Q ⊗ O b as O -algebras. Proposition 2.10
Let G be a finite group and let B be a block of O G with elementaryabelian defect group D of order and cyclic inertial quotient. Suppose N ✂ G with G = N D . If B covers a non-nilpotent G -stable block b of O N , then there is anelementary abelian -group Q ≤ D with G = N ⋊ Q such that B is Morita equivalentto a block C of O ( N × Q ) with defect group ( D ∩ N ) × Q ∼ = D . Proof . Let b D be a block of C G ( D ) with Brauer correspondent B , and write E = N G ( D, b D ) /C G ( D ) as described in the introduction. We may suppose G = N , so | E | ≤
7. By [41, Theorem 15] we have l ( B ) = | E | .Following [44], we may write D = D × D where D = C D ( N G ( D, b D )) and D =[ N G ( D, b D ) , D ]. We have D ⋊ E = D × ( D ⋊ E ). Since D ≤ N , E is cyclic and D is elementary abelian, we may choose Q to be a direct factor of D .By the main theorem of [44] there is a perfect isometry I : L K ( N G ( D, b D ) , b N G ( D,b D ) ) → L K ( G, B )such that I ( λ ∗ ζ ) = λ ∗ I ( ζ ) for all λ ∈ Irr( D ) and ζ ∈ L K ( N G ( D, b D ) , b N G ( D,b D ) ). Wehave N G ( D, b D ) ≤ C G ( D ) ≤ C G ( Q ). Let B ′ = ( b D ) C G ( Q ) . Now B ′ also has inertialquotient E and we may apply the same argument to obtain a perfect isometry J : L K ( N G ( D, b D ) , b N G ( D,b D ) ) → L K ( C G ( Q ) , B ′ )such that J ( λ ∗ ζ ) = λ ∗ J ( ζ ) for all λ ∈ Irr( D ) and ζ ∈ L K ( N G ( D, b D ) , b N G ( D,b D ) ). Wemay then apply Proposition 2.9 to I ◦ J − and the result follows. ✷ In the above note that if b is Morita equivalent to a block c of O M for some finitegroup M , then C is Morita equivalent to the block c ⊗ O Q of O ( M × Q ).6 emma 2.11 Let G be a finite group and N ✁ G with G/N of odd order (and solvable).Let B be a block of O G covering a G -stable block b of O N with defect group D ∼ = ( C ) .Suppose that B covers no nilpotent block of any normal subgroup M ✁ G with N ≤ M .If b is of type C × C or ( C ) , then B is also of one of these two types. Proof . It suffices to consider the case that [ G : N ] is an odd prime, say w . Notethat B and b share the defect group D .Suppose C G ( D ) = C N ( D ). Then the inertial quotient of B contains that of b withindex dividing w . Since C × C is maximal amongst subgroups of odd order of GL (2)and is the only subgroup containing C as a normal subgroup the result follows in thiscase.Suppose C G ( D ) = C N ( D ). Let ( D, b D ) be a b -subpair and let ( D, B D ) be a B -subpair with B D covering b D . If C G ( D ) N G ( D, b D ), then B D covers w con-jugates of b D and B D is the unique block of C G ( D ) covering b D , so N G ( D, B D ) = C G ( D ) N G ( D, b D ). Hence N G ( D, B D ) /C G ( D ) ∼ = N N ( D, b D ) /C N ( D ) and we are done inthis case. If C G ( D ) ≤ N G ( D, b D ), then N G ( D, B D ) ≤ N G ( D, b D ) as b D is the uniqueblock of C N ( D ) covered by B D . Now [ N G ( D, B D ) : C G ( D )] divides [ N N ( D, b D ) : C N ( D )] and we are done. ✷ An essential part of a reduction of Donovan’s conjecture to quasisimple groups isK¨ulshammer’s analysis in [27] of the situation of a normal subgroup containing thedefect groups of a block, which involves the study of crossed products of a basic alge-bra with an ℓ ′ -group. In the general setting he finds finiteness results for the possiblecrossed products, but in our situation we are able to precisely describe the possibilitiesusing knowledge of the Picard groups of certain basic algebras.Background on crossed products may be found in [27], but we summarize what weneed here. Let X be a finite group and R an O -algebra. A crossed product of R with X is an X -graded algebra Λ with identity component Λ = R such that each gradedcomponent Λ x , where x ∈ X , contains a unit u x . Given a choice of unit u x for each x ,we have maps α : X → Aut( R ) given by conjugation by u x and µ : X × X → U ( R )given by α x ◦ α y = ι µ ( x,y ) ◦ α xy , where U ( R ) is the group of units of R and ι µ ( x,y ) isconjugation by µ ( x, y ). The pair ( α, µ ) is called a parameter set of X in R . In [27] anisomorphism of crossed products respecting the grading is called a weak equivalence.By the discussion following Proposition 2 of [27] weak isomorphism classes of crossedproducts of R with X are in bijection with pairs consisting of an Out( R )-conjugacy classof homomorphisms X → Out( R ) for which the induced element in H ( X, U ( Z ( R )))vanishes and an element of H ( X, U ( Z ( R ))).Note that α : X → Aut( R ) restricts to a map X → Aut( Z ( R )), which makes Z ( R )an X -algebra. The k -algebras Z ( R ) /J ( Z ( R )) and U ( Z ( R ) /J ( Z ( R ))) also become X -algebras.Now suppose that X = h x i is a cyclic ℓ ′ -group. Following the strategy in [27,Section 3], U ( Z ( R )) ∼ = U ( Z ( R ) /J ( Z ( R ))) × (1 + J ( Z ( R )) and H ( X, U ( Z ( R ))) ∼ = H ( X, U ( Z ( R ) /J ( Z ( R )))) × H ( X, J ( Z ( R ))) .
7e have H ( X, J ( Z ( R ))) = 0 since X is an ℓ ′ -group. Now Z ( R ) /J ( Z ( R )) is acommutative semisimple k -algebra, which we denote A , and note as above that it is an X -algebra. Write A = A × · · · × A r , where each A i is a product of simple algebras con-stituting an X -orbit. We have H ( X, U ( A )) ∼ = H ( X, U ( A )) × · · · × H ( X, U ( A r )).We claim each H ( X, U ( A i )) vanishes. As a kX -module U ( A i ) is induced from thetrivial module of kY for some Y ≤ X , and so by Shapiro’s Lemma H ( X, U ( A i )) ∼ = H ( Y, k × ) (see [3, 2.8.4]), which vanishes since X is cyclic. We have shown that H ( X, U ( Z ( R ))) = 0 for each i .Now suppose that we have a finite group G and N ✁ G with G/N an ℓ ′ -group.Suppose that B is a block of O G covering a G -stable block b of O N . Define X = G/G [ b ]. Let C be a block of O G [ b ] covering b (and covered by B ), which by Proposition2.1 is Morita equivalent to b . Let f be an idempotent of C such that f Cf is a basicalgebra. Following [27], which is performed over O in [17], we may consider f Bf as acrossed product of f Cf with X , and f Bf is Morita equivalent to B .The Picard group Pic( R ) of R consists of isomorphism classes R - R -bimodules whichinduce Morita self-equivalences of b . For b - b -bimodules M and N , the group multipli-cation is given by M ⊗ b N . Let ϕ ∈ Aut( R ). Define the R - R -bimodule ϕ R by letting ϕ R = R as sets and defining a · m · a = ϕ ( a ) ma for a , a , m ∈ b . By [10, 55.11]inner automorphisms give isomorphic bimodules and ϕ ϕ R gives rise to an injectionOut( R ) → Pic( R ). If R is a basic algebra, then this is an isomorphism. We direct thereader to [4] for a thorough investigation of Picard groups of blocks with respect todiscrete valuation rings.For most of the blocks which appear as candidates for b in this paper the Picardgroup is known by [16]. We gather this information here: Proposition 3.1 ([16])
Let Q be a finite abelian -group.(i) Pic( O ( A × Q )) ∼ = S × ( Q ⋊ Aut( Q )) .(ii) Pic( B ( O ( A × Q ))) ∼ = C × ( Q ⋊ Aut( Q )) .(iii) Pic( O ( A × A )) ∼ = S ≀ C .(iv) Pic( B ( O ( A × A ))) ∼ = S × C . Applying all of the above, we have the following:
Proposition 3.2
Let G be a finite group and N ✁ G with G/N cyclic of odd primeorder. Let b be a G -stable block of O N with defect group D ∼ = ( C ) .(i) If b is Morita equivalent to O ( A × C × C ) , then B is Morita equivalent to b , O D , O ( A × A ) or a non-principal block of O ( C ) ⋊ , where the centre of acts trivially.(ii) If b is Morita equivalent to the principal block of O ( A × C × C ) , then B isMorita equivalent to b or the principal block of O ( A × A ) .(iii) If b is Morita equivalent to O ( A × A ) , then B is Morita equivalent to b , O ( A × C × C ) or O (( C ) ⋊ C )) where the C acts with only one fixed point.(iv) If b is Morita equivalent to the principal block of O ( A × A ) , then B is Moritaequivalent to b or the principal block of O ( C × C × A ) .(v) If b is Morita equivalent to the principal block of O ( A × A ) , then B is Moritaequivalent to b . roof . Let B be a block of G covering b . Either G [ b ] = G or G [ b ] = N . ByProposition 2.1 if G [ b ] = G , then B is Morita equivalent to b , in which case we aredone. Hence suppose G [ b ] = N , so by Proposition 2.1 B is the unique block of G covering b . Note that B and b share a defect group.We treat cases (i)-(iv) first. Case (v) uses a different strategy since we do not knowthe Picard group for the principal block of O ( A × A ).Define X = G/N . Consider an idempotent f of b such that R := f bf is a basicalgebra for b . Then we may consider Λ := f Bf as a crossed product of f bf with X and Λ is Morita equivalent to B . Note that Pic( b ) = Pic( R ).The weak equivalence classes of crossed products of R with X are in 1-1 corre-spondence with equivalence classes of homomorphisms α : X → Out( R ). Hence todetermine the possible Morita equivalence classes for B we must determine the equiv-alence classes of homomophisms α .Now if ker( α ) = X , then Λ ∼ = O X ⊗ f bf , Morita equivalent to a product of copiesof f bf . But this contradicts the fact that Λ is Morita equivalent to B . By Proposition3.1, in each case under consideration Pic( b ) has order divisible only by the primes 2and 3, and so we may suppose that | X | = 3 and that ker( α ) = 1.In many of the cases we will be making use of the example P SL (7) where there is ablock which is Morita equivalent to O A and covered by a nilpotent block of P GL (7).(i) Suppose b is Morita equivalent to O ( A × C × C ). By Proposition 3.1 wehave Pic( b ) ∼ = S × S and so there are three possibilities for α up to equivalence(recall that α is assumed to be faithful). The three possible Morita equivalence typesfor B are given by: O ( A × A ), realised when N is A × C × C ; O D , realisedwhen G = P GL (7) × C × C , N = P SL (7) × C × C ; a non-principal block of O ( C ) ⋊ , where the centre of 3 acts trivially, achieved when N is a maximalsubgroup of G = ( C ) ⋊ .(ii) Suppose b is Morita equivalent to the principal block of O ( A × C × C ). Wehave Pic( b ) ∼ = C × S and so there is just one possibility for α up to equivalence, andthis is achieved with G = A × A .(iii) Suppose b is Morita equivalent to O ( A × A ). We have Pic( b ) ∼ = S ≀ C .There are two non-trivial possibilities for α up to equivalence. They give rise to analgebra Morita equivalent to O ( A × C × C ), realised with G = A × P GL (7), and O (( C ) ⋊ C )) where the C acts with only one fixed point. The latter case is realisedwhen P SL (7) × P SL (7) = N < G < P GL (7) × P GL (7), with G the preimage ofthe diagonal subgroup of ( P GL (7) /P SL (7)) × ( P GL (7) /P SL (7)).(iv) Suppose b is Morita equivalent to the principal block of O ( A × A ). We havePic( b ) ∼ = S × C and so there is just one possibility for α up to equivalence, and thisis realised with G = P GL (7) × A .(v) Finally suppose b is Morita equivalent to the principal block of O ( A × A ).Then l ( b ) = 9 and b has a distinguished simple module identified by the unique columnof the decomposition matrix of b with all entries equal to 1, necessarily fixed under theconjugation action of G . Write w := | G/N | . Recalling that B is the unique block of G covering b , by Clifford theory we have: l ( B ) = 9 w if w ≥ l ( B ) ≥
15 if w = 7; l ( B ) ≥
21 if w = 5; l ( B ) ∈ { , , } if w = 3. By [29, Proposition 2.1] either l ( B ) ≤ l ( B ) = 15, this last case occuring when B has inertial quotient C . Hencewe are done unless possibly w = 7 and G acts with an orbit of length 7. However,9urther examination of the decomposition matrix for b reveals that there are preciselytwo columns with twelve non-zero entries, two with eight non-zero entries and fourwith four non-zero entries, so such an orbit of length seven is impossible. We haveexhausted all possibilities, so conclude that B must be Morita equivalent to b in thiscase. ✷ Corollary 3.3
Consider G = ( C ) ⋊ , where the centre of acts trivially. The -blocks of O G correspond to the simple modules of Z (3 ) , and the two non-principalblocks are Morita equivalent. Further, these blocks are Morita equivalent to the twonon-principal blocks of O (( C ) ⋊ − ) . Proof . Let B be any faithful 2-block of G = ( C ) ⋊ or ( C ) ⋊ − . Then l ( B ) = 1. Take a maximal subgroup N of G and a block b of N covered by B . Then N ∼ = (( C ) ⋊ C ) × C or ( C ) ⋊ C and b is Morita equivalent to O ( C × C × A ).By Proposition 3.2 there is only one possibility for the Morita equivalence class of B under the restriction that there is just one simple module. ✷ We first address the case where the defect group is normal.
Lemma 4.1
Let B be a block of O G for a finite group G with normal defect group D ∼ = ( C ) . Then B is Morita equivalent to a block as in (a) or (b)(i), (ii), (iv), (v),(vi), (viii), (xi) or (xiii) in Theorem 1.1. Proof . This follows from the main result of [26], applying Lemma 3.3 when theinertial quotient is C × C . ✷ For a block B , write IBr( B ) for the set of irreducible Brauer characters of B and l ( B ) = | IBr( B ) | .The following lemma deals for example with the situation SL n ( q ) ∼ = N ✁ G where G is an extension by field automorphisms and the block of SL n ( q ) is nilpotent covered. Lemma 4.2
Let G be a finite group and N ✁ G such that G/N is solvable. Let B be aquasiprimitive block of O G with abelian defect group D covering a block b of O N alsowith defect group D . If b is nilpotent covered, then B is Morita equivalent to a block ofa finite group with normal defect group. In particular, if D ∼ = ( C ) , then B is Moritaequivalent to one of the blocks in (a) or (b)(i), (ii), (iv), (v), (vi), (viii), (xi) or (xiii)of Theorem 1.1. Proof . By Proposition 2.6(ii) b is inertial, i.e., basic Morita equivalent to itsBrauer correspondent c in N N ( D ). Let M be the preimage in G of O ℓ ′ ( G/N ) and B M the unique block of M covered by B . By Proposition 2.6(iii) B M is inertial. Write10 for the preimage in G of O ℓ ( G/M ) and let B M be the unique block of M coveredby B . Note that B M and B M both have defect group D . Since M /M is an ℓ -group B M is the unique block of M covering B M . But then by [1, 15.1] M = M D , and so M = M . Since G/N is solvable this implies that M = G , and B is inertial. The lastpart follows by Lemma 4.1. ✷ We prove Theorem 1.1.
Proof . Let B be a block of O G for a finite group G with defect group D ∼ = ( C ) with ([ G : O ′ ( Z ( G ))] , | G | ) minimised in the lexicographic ordering such that B is notMorita equivalent to any of the sixteen blocks listed in the theorem.Suppose N ✁ G and b is a block of O N covered by B . Write I = I G ( b ) for thestabiliser of b under conjugation. Then there is a unique block B I of I covering b with Brauer correspondent B (the Fong-Reynolds correspondent) and B I is Moritaequivalent to B . Further B and B I share a defect group, hence by minimality I = G .Applying this to all normal subgroups of G , we have that B is quasiprimitive, that is,for every N ✁ G each block of O N covered by B is G -stable.By Corollary 2.3 and minimality, if N ✁ G and B covers a nilpotent block of O N ,then N ≤ Z ( G ) O ( G ). In particular O ′ ( G ) ≤ Z ( G ).Note that O ( G ) D = G . This holds by [1, 15.1] since any block of O ( G ) coveredby B is G -stable and B is the unique block of G covering it.Following [2] write E ( G ) for the layer of G , that is, the central product of thesubnormal quasisimple subgroups of G (the components ). Write F ( G ) for the Fittingsubgroup, which in our case is F ( G ) = Z ( G ) O ( G ). Write F ∗ ( G ) = F ( G ) E ( G ) ✁ G ,the generalised Fitting subgroup, and note that C G ( F ∗ ( G )) ≤ F ∗ ( G ). Let b ∗ be theunique block of O F ∗ ( G ) covered by B .We have E ( G ) = 1, since otherwise F ∗ ( G ) = F ( G ) = Z ( G ) O ( G ) and D ≤ C G ( F ∗ ( G )) ≤ F ∗ ( G ), so that D ✁ G , a contradiction by Lemma 4.1. Write E ( G ) = L ∗ · · · ∗ L t , where each L i is a component of G (we have shown that t ≥ B covers a block b E of O E ( G ) with defect group contained in D , and b E covers a block b i of O L i . Since b E is G -stable, for each i either L i ✁ G or L i is in a G -orbit in whicheach corresponding b i is isomorphic (with equal defect). Since B has defect four, itfollows that if t ≥
3, then B covers a nilpotent block of a normal subgroup generated bycomponents of G , a contradiction. Hence t ≤
2, and in particular
G/F ∗ ( G ) is solvableby the Schreier conjecture.We have | F ∗ ( G ) ∩ D | ≥
4, since otherwise B covers a nilpotent block of F ∗ ( G ), acontradiction since F ∗ ( G ) is not central in G .In the next part of the proof we will show that G (as a minimal counterexample) hasa proper normal subgroup N containing D such that the unique block b of N coveredby B is of type ( C ) or C × C .Suppose | F ∗ ( G ) ∩ D | = 4. Then F ∗ ( G ) ∩ D is normal in N G ( D ) and so anynon-nilpotent block of O ′ ( F ∗ ( G ) h D g : g ∈ G i ) has type ( C ) , ( C ) or C × C .We claim that O ′ ( F ∗ ( G ) h D g : g ∈ G i ) is a proper subgroup of G . For suppose O ′ ( F ∗ ( G ) h D g : g ∈ G i ) = G . Since O ( G ) = G , then F ∗ ( G ) h D g : g ∈ G i = G . Since G/F ∗ ( G ) is solvable it follows that O ( G ) = G . Since G = O ( G ) D , it follows that G has a normal subgroup H of index 2 containing F ∗ ( G ) such that G = HD . Hence B must have type ( C ) and we may apply Proposition 2.10 to show that by minimality11 is Morita equivalent to a block on the list. Hence O ′ ( F ∗ ( G ) h D g : g ∈ G i ) is aproper subgroup of G as claimed, and we take N = O ′ ( F ∗ ( G ) h D g : g ∈ G i ). As above O ( N ) = N and N has a normal subgroup of index 2, so that we may rule out thepossibilities that b has type ( C ) or C × C .Suppose that | F ∗ ( G ) ∩ D | = 8. It follows from Proposition 2.8 that one of moreof the following occurs: b ∗ is nilpotent covered; b ∗ has inertial quotient C ; or E ( G )is isomorphic to one of SL (8), G (3 m +1 ), J or Co . In the second case we maytake N = F ∗ ( G ) h D g : g ∈ G i and it is clear that b must be of type ( C ) . Since O ( G ) = G we have N = G . In the third case, each of the groups SL (8), G (3 m +1 ), J and Co has odd order outer automorphism group, so G has a direct factor of order2, contradicting O ( G ) = G . Suppose that b ∗ is nilpotent covered and does not haveinertial quotient C . By Proposition 2.6 we must have that b ∗ is inertial and Moritaequivalent to ( C ) ⋊ C . Now G/F ∗ ( G ) is solvable, so by Lemma 2.4 DF ∗ ( G ) /F ∗ ( G ) isa Sylow 2-subgroup of G/F ∗ ( G ), so that [ G : F ∗ ( G )] = 2. It follows that G/F ∗ ( G ) hasa normal 2-complement. Write M for the preimage in G of this normal 2-complementand write B M for the unique block of O M covered by B , so B M also covers b ∗ . ByProposition 2.6 B M is also inertial with abelian inertial quotient. Since B M has defectgroup ( C ) , this inertial quotient must then be cyclic, that is C or C . By [34,Corollary 3.7] G acts as inner automorphisms on B M , so in particular every simple B M -module is G -stable. Hence l ( B ) = l ( B M ) ∈ { , } and k ( B ) = 2 k ( B M ) = 16, sothat by [29] B also has cyclic inertial quotient. It follows by Proposition 2.10 that B is Morita equivalent to O ( D ⋊ C ) or O ( D ⋊ C ), contradicting minimality.Hence we may suppose that D ≤ F ∗ ( G ). We examine the possibilities for O ( G ).If | O ( G ) | = 16, then O ( G ) = D , a contradiction by Lemma 4.1. If | O ( G ) | = 8,then as E ( G ) = 1, B covers a nilpotent block of E ( G ), a contradiction. If | O ( G ) | = 4,then b ∗ must be of type ( C ) , ( C ) or C × C . However F ∗ ( G ) would have a normalsubgroup of index 2 and so we may rule out the cases of type ( C ) and C × C . If F ∗ ( G ) = G , then B is Morita equivalent to a block in the list, a contradiction. Hencewe may take N = F ∗ ( G ).Hence | O ( G ) | = 1 or 2, and O ( G ) ≤ Z ( G ).Suppose that t = 1. By Proposition 2.8 one or more of:(1) b ∗ has type ( C ) ; or(2) F ∗ ( G ) is isomorphic to one of C × SL (8), C × G (3 m +1 ), C × J , C × Co or SL (16), as in each of these cases the component must be simple (since the Schurmultiplier is trivial); or(3) b ∗ is nilpotent covered.In case (1) we may take N = F ∗ ( G ).Suppose case (2) occurs. If F ∗ ( G ) ∼ = C × J , C × Co or SL (16), thenOut( F ∗ ( G )) = 1 and so G = F ∗ ( G ). By [25] the non-principal block of Co withelementary abelian defect group of order 8 is Morita equivalent to the principalblock of Aut( SL (8)) and so in each of these three cases B is Morita equivalent to ablock in the list, a contradiction. If F ∗ ( G ) ∼ = C × SL (8), then G ∼ = C × SL (8)or C × Aut( SL (8)), again a contradiction. If F ∗ ( G ) ∼ = C × G (3 m +1 ), then G has C as a direct factor and by [12, 3.1] B is Morita equivalent to b ∗ . Hence byminimality G ∼ = C × G (3 m +1 ). By [36, Example 3.3], which in turn uses [30], b ∗ isMorita equivalent to the principal block of C × G (3) ∼ = C × Aut( SL (8)), again a12ontradiction to minimality.If (3) occurs, then we may apply Lemma 4.2 to obtain a contradiction.Now suppose that t = 2. Then b and b both have Klein four defect group and arenon-nilpotent, and b ∗ has type C × C . Hence we may take N = F ∗ ( G ).We have shown that there is a normal subgroup N ✁ G containing D and a block b of N covered by B with type ( C ) or C × C .Write J = G [ b ] ✁ G , and let B J be the unique block of O J covering b and coveredby B . By Proposition 2.1(i) B J is source algebra equivalent to b , and so in particular isalso of type ( C ) or C × C . Hence we may assume (repeatedly applying the argumentif necessary) that G [ b ] = N . Then by Proposition 2.1(ii) B is the unique block of G covering b . Hence by [1, 15.1] [ G : N ] is odd since B and b share a defect group, andso G/N is solvable (note that it is not strictly necessary to directly use the odd ordertheorem here, as in all the cases above N contains F ∗ ( G ) and G/F ∗ ( G ) is solvable).Let M ✁ G with [ G : M ] is an odd prime. Let B M be the unique block of M coveredby B . By Lemma 2.11 B M has type ( C ) or C × C .Now by minimality B M is Morita equivalent to a block as in (a), (b)(ii), (b)(iii),(b)(viii), (b)(ix) or (b)(x) in the statement of the theorem. Suppose that B M is as in(a). Then B M is inertial as by minimality there is only one possibility for the Moritaequivalence class of B M and of its Brauer correspondent in N M ( D ). So by Proposition2.6 B is also inertial and by Lemma 4.1 is Morita equivalent to one of the listed blocks,a contradiction. We now have that B M is Morita equivalent to one of the blocksconsidered in Proposition 3.2, which we apply to see that B is one of the blocks listedin the statement of the theorem.To see that the blocks in cases (a),(b) (i)-(xv) represent distinct Morita equivalenceclasses it suffices to note that the blocks in case (b) have distinct Cartan matrices andthe basic algebras for the blocks in (a) and (b)(i) are not isomorphic.That the blocks in case (a) cannot be Morita equivalent to a principal block fol-lows from [35, 6.13] that if the principal block has only one simple module, then it isnilpotent.Finally, we reference the literature that tells us that representatives of the Moritaequivalence classes with the same inertial quotient and number of simple modules arederived equivalent. In the cases below splendid derived equivalences are establishedbetween the relevant blocks defined with respect to k . Then by [39, 5.2] there is asplendid derived equivalence over O .By [39, §
3] the principal blocks of kA and kA are splendid derived equivalent. Itfollows that the blocks in cases (ii) and (iii) are derived equivalent, and that the blocksin cases (viii), (ix) and (x) are derived equivalent. The principal blocks of kSL (16)and k (( C ) ⋊ C ) (the normalizer of a Sylow 2-subgroup) are derived equivalentby [21], and so the blocks in cases (xi) and (xii) are splendid derived equivalent. Thatthe principal blocks of kJ and k (( C ) ⋊ ( C ⋊ C )) are derived equivalent followsfrom [20], and a published proof may be found in [9, § ✷ k , whichhelped me see their final structure and complete an earlier version of this paper. Ialso thank Markus Linckelmann for encouraging me to extend my results over k to O and for helpful discussions. The papers of Watanabe are essential in completing theclassification over O , and I am indebted to Hu Xueqin for directing me to [44] and toShigeo Koshitani for showing me [43] and [46]. Finally I thank Cesare Ardito for hiscareful reading of the manuscript and for his helpful comments, and Michael Liveseyfor some useful discussions. References [1] J. L. Alperin,
Local Representation Theory , Cambridge Studies in Advanced Math-ematics , Cambridge university Press (1986).[2] M. Aschbacher, Finite group theory , Cambridge Studies in Advanced Mathematics , Cambridge university Press (1986).[3] D. Benson, Representations and cohomology. I , Cambridge Studies in AdvancedMathematics , Cambridge University Press (1991).[4] R. Boltje, R. Kessar, and M. Linckelmann, On Picard groups of blocks of finitegroups , available arXiv:1805.08902[5] M. Brou´e and L. Puig,
Characters and local structure in G -algebras , J. Algebra (1980), 306–317.[6] M. Brou´e and L. Puig, A Frobenius theorem for blocks , Invent. Math. (1980),117–128.[7] D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann, The structure ofblocks with a Klein four defect group , Math. Z. (2011), 441–476.[8] D. A. Craven and A. Glesser,
Fusion systems on small p -groups , Trans. AMS (2012), 5945–5967.[9] D. A. Craven and R. Rouquier, Perverse equivalences and Brou´e’s conjecture , Adv.Math. (2013), 1–58.[10] C. W. Curtis and I. Reiner,
Methods of representation theory with applications tofinite groups and orders, Volumes I and II , John Wiley and Sons (1987).[11] E. C. Dade,
Block extensions , Ill. J. Math. (1973), 198-272.[12] C. W. Eaton, Morita equivalence classes of -blocks of defect three , Proc. AMS (2016), 1961–1970. 1413] C. W. Eaton, R. Kessar, B. K¨ulshammer and B. Sambale, 2 -blocks with abeliandefect groups , Adv. Math. (2014), 706-735.[14] C. W. Eaton, B. K¨ulshammer and B. Sambale, 2 -blocks with minimal nonabeliandefect groups, II , J. Group Theory (2012), 311–321.[15] C. W. Eaton and M. Livesey, Classifying blocks with abelian defect groups of rank for the prime
2, J. Algebra (2018), 1–18.[16] C. W. Eaton and M. Livesey,
Some examples of Picard groups of blocks , arXiv1810.10950[17] F. Eisele,
The Picard group of an order and K¨ulshammer reduction, availablearXiv:1807.05110.[18] K. Erdmann,
Blocks whose defect groups are Klein four groups: a correction , J.Algebra (1982), 505–518.[19] K. Erdmann, Blocks of tame representation type and related algebras , LectureNotes in Mathematics , Springer-Verlag (1990).[20] H. Gollan and T. Okuyama,
Derived equivalences for the smallest Janko group ,preprint (1997).[21] M. L. Holloway,
Derived equivalences for group algebras , Ph.D. thesis, Universityof Bristol (2001).[22] R. Kessar, S. Koshitani and M. Linckelmann,
Conjectures of Alperin and Brou´efor -blocks with elementary abelian defect groups of order
8, J. Reine Angew.Math. (2012), 85–130.[23] S. Koshitani,
Conjectures of Donovan and Puig for principal -blocks with abeliandefect groups , Comm. Alg. (2003), 2229-2243; Corrigendum , (2004), 391–393.[24] S. Koshitani and B. K¨ulshammer, A splitting theorem for blocks , Osaka J. Math. (1996), 343–346.[25] S. Koshitani, J. M¨uller and F. Noeske, Brou´e’s abelian defect group conjectureholds for the sporadic simple Conway group Co , J. Algebra (2011), 354–380.[26] B. K¨ulshammer, Crossed products and blocks with normal defect groups , Comm.Alg. (1985), 147–168.[27] B. K¨ulshammer, Donovan’s conjecture, crossed products and algebraic group ac-tions , Israel J. Math. (1995), 295–306.[28] B. K¨ulshammer and L. Puig, Extensions of nilpotent blocks , Invent. Math. (1990), 17–71.[29] B. K¨ulshammer and B. Sambale,
The -blocks of defect
4, Representation Theory (2013), 226–236. 1530] P. Landrock and G. Michler, Principal -blocks of the simple groups of Ree type ,Trans. Amer. Math. Soc. (1980), 83–111.[31] P. Landrock and B. Sambale, On centers of blocks with one simple module , J.Algebra (2017), 339–368.[32] M. Linckelmann,
The source algebras of blocks with a Klein four defect group , J.Algebra (1994), 821–854.[33] M. Linckelmann,
The isomorphism problem for cyclic blocks and their source al-gebras , Invent. Math. (1996), 265–283.[34] M. Murai,
On blocks of normal subgroups of finite groups , Osaka J. Math. (2013), 1007–1020.[35] G. Navarro, Characters and blocks of finite groups , London Mathematical SocietyLecture Note Series , Cambridge University Press (1998).[36] T. Okuyama,
Some examples of derived equivalent blocks of finite group , preprint(1997).[37] L. Puig,
Nilpotent blocks and their source algebras , Invent. Math. (1988), 77–116.[38] L. Puig, Nilpotent extensions of blocks , Math. Z. (2011), 115-136.[39] J. Rickard,
Splendid equivalences: derived categories and permutation modules ,Proc. London Math. Soc. (1996), 331–358.[40] B. Sambale, 2 -blocks with minimal nonabelian defect groups , J. Algebra (2011), 261–284.[41] B. Sambale, Cartan matrices and Brauer’s k ( B ) -conjecture IV , J. Math. Soc.Japan (2017), 735–754.[42] A. Watanabe, On nilpotent blocks of finite groups , J. Algebra (1994), 128–134.[43] A. Watanabe,
A remark on a splitting theorem for blocks with abelian defect groups ,RIMS Kokyuroku Vol.1140, Edited by H.Sasaki, Research Institute for Mathemat-ical Sciences, Kyoto University (2000) 76–79.[44] A. Watanabe,
On perfect isometries for blocks with abelian defect groups and cyclichyperfocal subgroups , Kumamoto J. Math. (2005), 85-92.[45] C. Wu, K. Zhang and Y. Zhou, Blocks with defect group Z n × Z n × Z m , J.Algebra (2018), 469–498.[46] Y. Zhou, On the p ′ -extensions of inertial blocks , Proc. AMS144