Morita equivalence classes of 2‐blocks with abelian defect groups of rank 4

We classify all 2‐blocks with abelian defect groups of rank 4 up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application is that Broué's abelian defect group conjecture holds for all blocks under consideration here.


Introduction
Let p be a prime and (K, O, k) be a p-modular system with k algebraically closed, and let P be a finite p-group.It is known by work culminating in [21] that if P is an abelian 2-group, then there are only finitely many Morita equivalence classes amongst all blocks of OG for all finite groups G with defect group D isomorphic to P , i.e., that Donovan's conjecture holds for such P .This suggests the problem of classifying the Morita equivalence classes of blocks that arise for given abelian 2-groups P .The Morita equivalence classes are already known without the use of the classification when P is abelian and Aut(P ) is a 2-group and when P is a Klein four group.In the former case every block must be nilpotent by [49] since P controls fusion for the block, and in the latter there are three Morita equivalence classes by [40] (in fact these are the only source algebra equivalence classes by [14]).In [22] the classification of finite simple groups was applied to describe the 2-blocks of quasisimple groups with abelian defect groups.This has been used to classify the Morita equivalence classes of blocks with defect groups isomorphic to P in the following cases.When P is elementary abelian of order 16 or 32, there are 16 or 34 Morita equivalence classes by respectively [19] and [3].When P is elementary abelian of order 64, there are 81 Morita equivalence classes containing Inertial quotient Notes Table 1: Possible inertial quotients a principal block by [4].Morita equivalence classes of blocks when P is an abelian 2group of rank at most three are determined in [24] and [56].Further classifications are obtained when we place restrictions on the inertial quotient of the blocks we consider, as in [47] and [5], where it is assumed that the inertial quotient contains an element of maximal order, or is a subgroup of a cyclic group of maximal order.Note that for odd primes, full classifications of Morita equivalence classes of blocks with abelian defect groups are currently only known when P is a cyclic 3-group or C 5 .Morita equivalence classes have also been determined for some classes of nonabelian defect groups, which for completeness we briefly list here.Of the tame blocks with nonabelian defect groups, complete classifications are only known for dihedral 2-groups (over k, for which see [29] and [44]) and Q 8 (over O, for which see [28]).Aside from p-groups with only one saturated fusion system (for which blocks must be nilpotent), Morita equivalence classes of blocks with respect to O have been determined when P is a Suzuki 2-group (see [20]), an extraspecial p-group of order p 3 and exponent p for p ≥ 5 (see [2]), or is in a class of minimal nonabelian 2-groups as in [23].
Here we determine the Morita equivalence classes of blocks when p = 2 and P is abelian of rank at most four.There are two main problems to overcome in the reduction to quasisimple groups in order to apply [22]: the cases of a normal subgroup of odd index and of a normal subgroup of index two.In the former case we use Picard groups and crossed products to deduce possible Morita equivalence classes for the block of the over group.In the latter we use a combination of a method developed in [56] and one of our own. Let , where n 1 , . . ., n 4 ≥ 0. The conjugacy classes of odd order subgroups E of Aut(D) have representatives as given in Table 1, and correspond to the possible inertial quotients for blocks with defect group D (where by inertial quotient, defined later, we consider also the action on D).By [31, 5.2.3] we may write D = [D, E]×C D (E).To simplify notation we assume that labeling is chosen so that for some i.If B is a block with defect group D and inertial quotient E, then we say that B is of type E, where we use the notation of Table 1 to distinguish between isomorphic but nonconjugate subgroups of Aut(D).Note that the case (C 3 ) 2 represents the simultaneous action of C 3 on C 2 n 1 × C 2 n 2 and on C 2 n 3 × C 2 n 4 , which for n 1 = n 2 = n 3 = n 4 , also represents a subgroup of C 15 .
We now give the classification.Throughout this paper, G n will denote the non-abelian group (C Following [51] a block is called inertial if it is basic Morita equivalent to a block with normal defect group. Theorem 1.1.Let G be a finite group and B be a block of OG with defect group D ∼ = C 2 n 1 × C 2 n 2 × C 2 n 3 × C 2 n 4 as above.Let E be an inertial quotient of B, with notation as in Table 1.(iv) If E acts as C 15 , then either B is inertial or B is basic Morita equivalent to B 0 (OSL 2 (16)).(ii) Implicit in the statement of Theorem 1.1 is the assertion that for these defect groups Morita equivalent blocks have the same fusion (i.e., the same inertial quotient).
A consequence is the following, that Broué's abelian defect group conjecture holds for the blocks under consideration: Corollary 1.3.Let G be a finite group and B be a 2-block of OG with abelian defect group D of rank at most four.Let b be the Brauer correspondent of B in ON G (D). Then B is derived equivalent to b.Since Broué's conjecture is known for blocks with elementary abelian defect groups of order 32 by [6], we immediately have that the conjecture holds for all blocks with abelian defect groups of order dividing 32.
Remark 1.4.Broué's abelian defect group conjecture is often stated as requiring further a splendid Rickard equivalence.Since Theorem 1.1 only provides a Morita equivalence with no additional information on the bimodule affording the equivalence, we make no claim to the existence of a splendid equivalence in our result.
The structure of the paper is as follows.In Section 2 we briefly cover necessary notation and general results.This includes: results on covering of blocks of normal subgroups and related Morita equivalences; subpairs and inertial quotients; a review of the different (stronger) versions of Morita equivalence that we use, and their relationship with fusion; a summary of the results of [22] that we use; and Picard groups.We prove a result on extending Morita equivalences (in certain cases) from normal subgroups of index 2 in Section 4. This requires knowledge of some groups of perfect self-isometries, which is the content of Section 3. In Section 5 we give results on extending Morita equivalences from normal subgroups, which includes extensions using crossed products where the index is odd, combined with techniques for index 2 from the previous section and the result developed in [56].In Section 6 we present the proof of the main result.Finally in Section 7 we prove Corollary 1.3.

Notation and background 2.1 Morita equivalence
See [41, 2.8] for an introduction to Morita theory.It is not known whether Morita equivalence of blocks of finite groups defined over O preserve the isomorphism type of the defect group (Morita equivalences defined over k are known not to necessarily preserve this by [30]).However, by [10] Morita equivalence does preserve the isomorphism class of abelian defect groups.It is also not known whether the inertial quotient (and its action) are preserved.We will have to take some care in the proof of Theorem 1.1 to check that this does happen in our situation.
A Morita equivalence is basic if it is induced by an endopermutation source bimodule, and a block is inertial if there is basic Morita equivalence with the Brauer correspondent block of the normalizer of a defect group (see [50] and [51]).A basic Morita equivalence preserves the isomorphism type of the defect group and fusion.
We will also refer to source algebra equivalences (see [42, 6.4]).These occur in the section on Picard groups.Also certain Morita equivalences that we use are stated as source algebra equivalences (for example Lemma 2.2), but we do not use properties beyond that they are basic Morita equivalences.
Another point to mention here is the Bonnafé-Rouquier correspondence as in [12].Whilst correspondent blocks are only known to be Morita equivalences (i.e, not basic), they are splendid Rickard equivalent, which implies they have the same fusion.

Blocks and normal subgroups
We collect some background on blocks and normal subgroups that will be used frequently and usually without further reference.A reference for this material is [42,Section 6.8] Let G be a finite group and B be a block of OG with defect group D. We use prj(B) to denote the set of characters of projective indecomposable B-modules.Note that the version of the first part of Proposition 2.1 over k is also found within [34].
Recall that a block B is quasiprimitive if every block of every normal subgroup covered by B is G-stable.In particular B covers a unique block for each normal subgroup.
We will make frequent use of the following, especially in the case where the quotient group is a subgroup of the outer automorphism group of a quasisimple group.Lemma 2.2 (Lemma 2.4 of [3]).Let G be a finite group and let N ✁ G with G/N solvable.Let b be a G-stable block of ON and let B be a quasiprimitive block of OG covering b with defect group D. Then DN/N is a Sylow p-subgroup of G/N.
The normal subgroup G[b] of G is defined to be the group of elements of G acting as inner automorphisms on b ⊗ O k (see [37]).

Subpairs, fusion and inertial quotients
Let G be a finite group and B be a block of OG with defect group D. For convenience in stating definitions we assume that The B-subpairs define a fusion system F = F B (D, B D ), often called the Frobenius category of the B (see for example [8] or [41]).The inertial quotient of B is E = N G (D, B D )/DC G (D), together with the action of E on D. Since D is abelian, F is determined by E. Consequently, every block with abelian defect groups and trivial inertial quotient is nilpotent (we will use this fact throughout this paper without further reference).
The following is well-known.Most of the following is Lemma 5.
Proof.That (Q, b Q ) is a b-subpair forms part of the proof of the main theorem of [34].We note that due to the correspondence between O-blocks and k-blocks, the result may be proved for k-blocks, which is the setting for [34].Note also that this part of the proof of the main result of [34] does not require that G is a split extension of N.
We have Since N G (D, B D ) controls fusion in D, by [1,Proposition 4.24] we have The next result is essentially extracted from the proof of [56,Lemma 6.3], and allows us to compare inertial quotients of blocks with those of blocks of normal subgroups of p ′ index.We include the proof here for convenience.
and the result follows.

Blocks of quasisimple groups
We extract the results of [22] necessary for this paper.Recall that a block B of a group G is nilpotent covered if there is H with G ✁ H and a nilpotent block of H covering B. Properties of such blocks are considered in [51].In particular nilpotent covered blocks are inertial by [51,Corollary 4.3].

Proposition 2.7 ([22]
).Let p = 2, and let B be a block of OG for a quasisimple group G with abelian defect group D of rank at most 4. Then one or more of the following occurs: , where q = 3 2m+1 for some m ∈ N, and B is the principal block; (ii) G ∼ = Co 3 and B is the unique non-principal 2-block of defect 3; (iii) B has inertial quotient of type (C 3 ) 1 and is Morita equivalent to a block C of OL where L = L 0 × L 1 ≤ G such that L 0 is abelian and the block of OL 1 covered by C has Klein four defect groups; (iv) B is nilpotent covered.
Proof.This follows from Theorem 6.1 of [22] and its proof.The only point to address is the inertial quotient in case (iii).Case (iii) arises in two ways.The first is where in which case either B is nilpotent (and so is covered by case (iv)), or has inertial quotient C 3 .The second way this arises is as in case (v) of [22,Proposition 5.3].
Here the given Morita equivalence is given by the Bonnafé-Rouquier correspondence as in [12].Now the corresponding blocks in [12] are equivalent by a splendid Rickard equivalence and so have the same fusion.Hence B has the same inertial quotient as C, which is of type (C 3 ) 1 .
Corollary 2.8.Every 2-block of a quasisimple group that is Morita equivalent to G n , for n ∈ N, is inertial.
Proof.For n ≥ 2 we see from Proposition 2.7 that blocks of quasisimple groups with defect groups (C 2 n ) 2 are nilpotent covered, and so inertial.For n = 1, by [14] every block that is Morita equivalent to OA 4 is source algebra equivalent to OA 4 , and so inertial.

Picard groups
Let G be a finite group and B be a block of OG with defect group D. The Picard group Pic(B) of B has elements the isomorphism classes of B-B-bimodules inducing O-linear Morita auto-equivalences of B. For B-B-bimodules M and N, the group multiplication is given by M ⊗ B N. For background and definitions we follow [11].
We will use knowledge of Pic(B) to refine Külshammer's analysis in [38] of the situation of a normal subgroup containing the defect groups of a block.This involves the study of crossed products of a basic algebra with a p ′ -group, which in turn uses the outer automorphism group of the basic algebra, a group that embeds into the Picard group.This will be essential in reduction steps in our classification of Morita equivalence classes of blocks.
Write T (B) for the subgroup of Pic(B) consisting of bimodules with trivial source and L(B) for the subgroup consisting of linear source modules.
T (B) and L(B) are described in [11], and we summarise the relevant notation and results here.Let F = F B (D, B D ) be the fusion system for B on D, defined using a B-subpair (D, B D ), and let Let A be a source algebra for B. Then A is an interior D-algebra, so we have an embedding of D into A and we may consider the fixed points A D under the action of D. Write Aut D (A) for the group of O-algebra automorphisms of A fixing each element of the image of D in A, and Out D (A) for the quotient of Aut D (A) by the subgroup of automorphisms given by conjugation by elements of (A D ) × .By [49, 14.9] Out D (A) is isomorphic to a subgroup of Hom(E, k × ).
By [11,Theorem 1.1] we have exact sequences where foc(D) is the focal subgroup of D with respect to F , generated by the elements ϕ We record for later use that by [52], if P is a p-group, then We gather together here the results regarding Picard groups that we will use later on.
given by those self-equivalences fixing the projective indecomposable modules is isomorphic to P ⋊ Aut(P ).
Proof.Throughout this proof, we denote by D a defect group of the block in question.
(i) The Picard group is described in [27,Theorem 1.1].The S 3 factor consists of (bimodules inducing) Morita equivalences corresponding to elements of where the C 3 is generated by multiplying by a non-trivial linear character of G n .The P ⋊ Aut(P ) factor is generated by equivalences given by multiplication by a (linear) character and by automorphisms of P .Since the projective indecomposable modules correspond in this case to the irreducible characters with D in their kernel, the remainder follows. (ii , where E is the inertial quotient (note that OG has a unique block).By [43]  Remark 2.10.In case (v), whilst a Sylow 3-subgroup of the Picard group for the block in question occurs as a conjugate of Out(D, F ), this is not known to be the case for every block Morita equivalent to it.In other words, it is theoretically possible for the Picard group of a Morita equivalent block C to have a subgroup C 7 ⋊ C 3 , but for T (C) ∼ = C 7 ⋊ C 3 .We will have to beware of this inconvenience in Section 5.

Preliminaries on perfect isometries
We require a method for comparing the principal blocks of O(A 4 × P ) and O(A 5 × P ) with those of O(A 4 ×Q) and O(A 5 ×Q) respectively when Q is a subgroup of an abelian 2-group P .We will do this in Section 4, but first require an analysis of their perfect self-isometries, which is the content of this section.For a block B, write Perf(B) for the group of perfect self-isometries of B, under composition.The results of this section are an extension of those of part of Section 2 of [24].
Note that every perfect isometry I between blocks B 1 and B 2 gives rise to a bijection of character idempotents and so to a K-algebra isomorphism between Z(KB 1 ) and Z(KB 2 ), and that by [13] this induces an O-algebra isomorphism φ I : Z(B 1 ) → Z(B 2 ).
By Proposition 2.1, for a block with abelian defect groups, every irreducible character in a block of a normal subgroup of index p covered by B is G-stable and extends to G. The following Proposition tells us that these extensions behave well with respect to perfect isometries.Proposition 3.1.For i = 1, 2 let G i be a finite group and N i ✁ G i with index p.Let B i be a block of OG i with abelian defect group D and let b i be a G i -stable block of ON i covered by B i .For each χ ∈ Irr(b 1 ) write Irr(B 1 , χ) = {χ 1 , . . ., χ p }.
Suppose I : Z Irr(B 1 ) → Z Irr(B 2 ) is a perfect isometry such that for each χ ∈ Irr(b 1 ) there is ψ ∈ Irr(b 2 ) and ǫ χ ∈ {±1} such that I(χ i ) = ǫ χ ψ i for i = 1, . . ., p where Irr(B 2 , ψ) = {ψ 1 , . . ., ψ p }. Then the isometry Remark 3.2.We may restrict φ I to Z(b 1 ) since, as by Proposition 2.1 B i acts as inner automorphisms on b i and so Proof.This is Proposition 2.6 and Lemma 2.7 of [24].Lemma 3.3.Let P be an abelian finite p-group.Then Perf(OP ) Proof.Since there is only one indecomposable projective module for OP , every perfect self-isometry of OP must have all positive or all negative signs.Now every perfect self-isometry induces a permutation of Irr(P ), which induces an automorphism of Aut(Z(OP )) ∼ = Aut(OP ), and the result follows.Now consider the character table of A 4 .Let ω be a primitive 3rd root of unity.We set up the labelling of characters: For the rest of the section we assume p = 2.
Let I be a perfect self-isometry of O(P × A 4 ).Each I(χ P i ) is an integer linear combination of projective indecomposable characters.By counting constituents we see that for 1 ≤ u ≤ 3. Consider the set we see that in fact all these signs are the same and we may assume, after possibly composing I with (Id Z Irr(P ) , I 1,−1 ), that for each θ ∈ Irr(P ).As 3 is invertible in O, this implies where δ m is defined as in Proposition 3.4, and so for each θ ∈ Irr(P ).Fixing for now θ ∈ Irr(P ), define θ m ⊗ χ m := I(θ ⊗ χ m ), for 1 ≤ m ≤ 4. Let x ∈ P .Evaluating (3) at (x, 1), (x, (123)) and (x, (132)), gives Proceeding as in the proof of [24,Theorem 2.11] we have for all x ∈ P .In other words We have shown that we may assume I is of the form for all θ ∈ Irr(P ), where J is a permutation of Irr(P ).In particular the O-algebra automorphism of Z(O(P × A 4 ) induced by I leaves OP invariant.Therefore the permutation J of Irr(P ) must induce an automorphism of OP and the theorem is proved.
We need two technical lemmas before we continue.Set Proof.Throughout this proof we use X 1 , . . ., X s , X, Y, Z to denote the images of the elements of the same name in A (m 1 ,...,ms) ⊗ k A.
We first note that forms a basis for A (n 1 ,...,nt) ⊗ k A and, setting J := J(A (m 1 ,...,ms) ⊗ k A) (the Jacobson radical of A (m 1 ,...,ms) ⊗ k A), we have that Then W has order of nilpotency at least 2. If λ i = 0 for some 1 ≤ i ≤ s, then, by looking at the coefficients of powers of X i with respect to the basis B, we can see that W has order of nilpotency at least 2 m i .Now let W 1 , . . .W s+3 ∈ J be such that {W 1 + J 2 , . . ., W s+3 + J 2 } forms a basis of J/J 2 .We set o i to be the order of nilpotency of W i , for 1 ≤ i ≤ s + 3.By reordering, we may assume that these lower bounds on the o i 's can all be achieved.We have now shown that the tuple (m 1 , . . ., m s ) can be retrieved from the isomorphism type of A (m 1 ,...,ms) ⊗ k A and the result is proved.
We may assume that P, Q = 1.Note that Z(kA 4 ) ∼ = A, kP ∼ = A (m 1 ,...,ms) and kQ ∼ = A (n 1 ,...,nt) , where have isomorphic inertial quotients (C 3 ) 1 .Since l(B) is equal to the order of the inertial quotient, by the main result of [55] and its proof we have a perfect isometry Similarly there is a perfect isometry Z Irr(B 0 (O(P × A 5 ))) → Z Irr(O(P × A 4 )).Write A for A 4 or A 5 .From the above we have a perfect isometry where sgn G N is the linear character of G with kernel N.So for each θ ∈ Irr(b), L swaps the two extensions of θ to G. We know that L is indeed a perfect isometry as it is induced by the O-algebra automorphism of OG given by g → sgn G N (g)g for all g ∈ G.Note that L is a perfect self-isometry of order 2 and that it induces the trivial k-algebra automorphism on Z(kB).Furthermore, since by Proposition 2.5 induction gives a bijection between prj(b) and prj(B), each element of prj(B) is fixed under multiplication by sgn G N and so L is the identity on Zprj(B).Therefore I PI (L) must be of order 2, induce the identity k-algebra automorphism on Z(B 0 (k(P ×A))) and be the identity on Zprj(B 0 (O(P × A))).We claim that I PI (L) is induced by multiplication by the sign character of G ′ := P × A with respect to the subgroup N ′ := R × A ≤ G ′ , for some index 2 subgroup R ≤ P , with R ∼ = Q.
We first deal with the A = A 4 case.Adopting the notation of Theorem 3.5, set I PI (L) = (J, I σ,ǫ ), where J is a perfect self-isometry of OP induced by an O-algebra automorphism, σ ∈ S 4 and ǫ ∈ {±1}.Then the fact that I PI (L) is the identity on Zprj(O(P × A 4 )) forces σ to be the identity permutation and ǫ = 1.So J is induced by an element of α ∈ Aut(OP ) that has order 2 and induces the identity on kP .Recall that Aut(OP ) ∼ = Hom(P, O × ) ⋊ Aut(P ).The fact that α induces the identity on kP forces α to be given by multiplication by λ α ∈ Hom(P, O × ) of order 2. In other words α is induced by multiplication by the sign character of P with respect to some normal subgroup R of index 2. Hence I PI (L) is induced by multiplication by the sign character of P × A 4 with respect to the subgroup N ′ := R × A 4 ≤ G ′ .(Note that we do not know yet that R ∼ = Q.) We have now shown that for all χ ∈ Irr(B).By Proposition 3.1, b is then perfectly isometric to ON ′ .However, b is Morita equivalent to B 0 (O(Q × A 4 )) and so Corollary 3.7 implies that R ∼ = Q as desired.
For the A = A 5 case we fix a perfect isometry I A : Z Irr(B 0 (OA 5 )) → Z Irr(OA 4 ).As above, we can then show that Let a ∈ B be a graded unit as described in Proposition 4.1 and set a ′ := φ I (a).Since φ I respects the G/N and G ′ /N ′ -gradings, a ′ is also a graded unit.We now give M the structure of a module for by defining a ′−1 .m.a = m, for all m ∈ M, where (7) ensures that this does indeed define a module.Now by [45,Theorem 3.4] we have proved that B is Morita equivalent to B 0 (O(P × A)).

Extensions of blocks
In this section we give the possible Morita equivalence classes of blocks covering a block of a normal subgroup in some relevant Morita equivalence classes.
Let G be a finite group and N ✁ G. Let b be a G-stable block of ON covered by a block B of G with abelian defect group D. Then b has defect group Q = D ∩ N. Let (D, B D ) be a maximal B-subpair.
We first extract and summarize two results of [56] and [57].
The following is a weaker version of Theorem 5.10 of [56] that is sufficient for our purposes.The following is the main result of [57], and is particularly relevant to the case that b is a nilpotent covered block.

Theorem 5.2 ([57]
). Suppose that N has p ′ -index.If b is inertial, then B is inertial.
We now apply Külshammer's analysis in [38] of the situation of a normal subgroup containing the defect groups of a block, which involves the study of crossed products of a basic algebra with an p ′ -group.In the general setting he finds finiteness results for the possible crossed products, but in our situation we are able to precisely describe the possibilities.
Background on crossed products may be found in [38], but we summarize what we need 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 1 = R such that each graded component 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) is conjugation by µ(x, y).The pair (α, µ) is called a parameter set of X in R. In [38] an isomorphism of crossed products respecting the grading is called a weak equivalence.By the discussion following Proposition 2 of [38] weak isomorphism classes of crossed products of R with X are in bijection with pairs consisting of an Out(R)-conjugacy class of homomorphisms X → Out(R) for which the induced element in H 3 (X, U(Z(R))) vanishes, and an element of H 2 (X, U(Z(R))).
We adapt Proposition in Section 3 of [38], and include a proof for completeness (as given in [19]).Note that α : X → Aut(R) restricts to a map X → Aut(Z(R)).Hence we also have homomorphisms X → Aut(Z(R)/J(Z(R))) and ) is a commutative semisimple k-algebra, which we denote by A, and note as above we have a homomorphism X → Aut(A).Write A = A 1 × • • • × A r , where each A j is a product of simple algebras constituting an X-orbit.
We have vanishes, for as a kX-module A j is induced from the trivial module of kY for some Y ≤ X, and so by Shapiro's Lemma H i (X, U(A j )) ∼ = H i (Y, k × ) (see [9, 2.8.4]), which vanishes since X is cyclic.Hence H i (X, U(Z(R))) = 0 for each i.
We apply the above with X = G/N = x , where G/N is a p ′ -group.Let f be an idempotent of b such that f bf is a basic algebra for b.
f Bf is a crossed product of f bf with X and f Bf is Morita equivalent to B. Hence we may take R = f bf in the above.By Lemma 5.3 weak isomorphism classes of crossed products of f bf with X are in bijection with Out(f bf )-conjugacy classes of homomorphisms X → Out(f bf ).Note however, that such crossed products may be isomorphic as algebras but not weakly isomorphic as crossed products.Indeed, given α : X → Out(f bf ), the same algebra gives rise to parameter sets associated to α • ϕ for each ϕ ∈ Aut(X).Now Out(f bf ) embeds in Pic(f bf ) ∼ = Pic(b) and since f bf is a basic algebra Out(f bf ) ∼ = Pic(f bf ), so we may apply the descriptions of Picard groups in Proposition 2.9.The strategy will be to limit the number of possible Morita equivalence classes for B (given b), and to identify examples where all such Morita equivalence classes are realised.
A special case that will arise frequently, and that demonstrates the phenomenon of crossed products isomorphic as algebras but not weakly isomorphic as crossed products is as follows: Example 5.5.As observed in [3], there are examples of nilpotent blocks covering nonnilpotent blocks constructed in [51,Remark 4.4] that will be useful in the arguments that follow.Let P be an abelian 2-group on which C r acts regularly for a prime r.Define N = (P ⋊ C r ) × C r with Sylow r-subgroup H.Note that N has r 2-blocks, corresponding to Irr(Z(N)).There is a group T ∼ = C r acting on N fixing the elements of Z(N) and H/Z(N).Define G = N ⋊T .Then each nontrivial element of Irr(Z(N)) is covered by a nilpotent block of G (with defect group P ).These cover block of N Morita equivalent to O(P ⋊ C r ).We in particular require the cases Remark 5.6.In the crossed product construction, we are considering homomorphisms G/N → Pic(b) induced by congugation by elements of G. Conjugation on ON affords a permutation module, and so conjugation on an invariant summand (in our case b) affords a trivial source module.It follows that actually we have G/N → T (b).In general this is a less useful observation than we might hope, as T (b) is not (known to be) preserved under Morita equivalence.However, it does allow us to keep some control over inertial quotients as we move from b to B. We will use this observation in the following proofs.
We now apply the above to every extension of a Morita equivalence from a normal subgroup of odd prime index that we will need.
Further, if b is known to have inertial quotient (C 3 ) 1 , then in the latter two cases the inertial quotient of B is C 3 × C 3 .If r = 3 and P is cyclic or a product of two cyclic groups of different orders, then B is either nilpotent or source algebra equivalent to b.
(iii) Suppose that b is Morita equivalent to B 0 (O(A 5 × P )) where P is an abelian 2group with rank at most 2. If P is cyclic or a product of two cyclic groups of different orders, then B is source algebra equivalent to b.
, with the latter case only occurring when r = 3.
)) or source algebra equivalent to b, with the former case only occurring when r = 3.
Proof.(i) By Proposition 2.9 Pic(b . By Lemma 5.4, for both r = 3 and r = 7 there are two possibilities for the Morita equivalence class of B, and one is that B is source algebra equivalent to b.For r = 7 the second is that B is nilpotent as realised in Example 5.5.For r = 3, the second case is realised by the group given in the statement.For all other odd primes, since they do not divide the order of the Picard group, B is source algebra equivalent to b.It remains to prove the statement regarding inertial quotients.Suppose that b is known to have inertial quotient C (ii) By Proposition 2.9 Pic(b) ∼ = S 3 ×(P ⋊Aut(P )).If P is cyclic or a product of two cyclic groups of different orders, then Aut(P ) is a 2-group, and so by Lemma 5.4 B is either Morita equivalent to b or is nilpotent, with this case realised as in Example 5.5.
Suppose that P ∼ = (C 2 n ) for some n ∈ N. Then Pic(b) ∼ = S 3 × (P ⋊ S 3 ).There are four conjugacy classes of homomorphisms G/N → Pic(b), and so at most four possibilities for the Morita equivalence class of B. These are account for by the four cases listed in the statement, with the cases that B is Morita equivalent to b or O(G m ×G n ) requiring no further explanation.The case that B is nilpotent is again realised as in Example 5.5.Note that D ⋊ 3 1+2 + has a normal subgroup M of index 3 isomorphic to N × C 3 .Now B covers a nonprincipal block of OM, which is Morita equivalent to ON. Hence the final case is realised.It remains to prove the statement regarding inertial quotients.Suppose that b is known to have inertial quotient (C 3 ) 1 .It follows from Lemma 2.6 that B has inertial quotient (C 3 ) 1 or C 3 × C 3 .as above, by Remark 5.6 we observe that we may consider elements of T (b).By the description of T (b) in Section 2.5 we know that T (b) maps to a subgroup of Out(D, F ) ∼ = C 3 .If this subgroup is trivial, then we may not construct any nontrivial homomorphism G/N → T (b) and we are done.Hence suppose the subgroup has order three.Then the action of G induces an element of order three in Aut(D, F ), so that B must have inertial quotient C 3 ⋊ C 3 .
(iii) By Proposition 2.9 Pic(b) ∼ = C 2 × (P ⋊ Aut(P )).If P is cyclic or a product of two cyclic groups of different orders, then Pic(b) is a 2-group, and so B is Morita equivalent to b. Suppose that P ∼ = (C 2 n ) for some n ∈ N. Then Pic(b) ∼ = C 2 × (P ⋊ S 3 ).Hence by Lemma 5.4 the result follows.
(iv) By Proposition 2.9 Pic(b) ∼ = C 2 × S 3 .By Lemma 5.4 there are two possibilities for the Morita equivalence class of B, one of which is the class containing b.The case ) is realised by taking a product of A 5 with a group as in Example 5.5.
Remark 5.8.Since (a) source algebra equivalence preserves the inertial quotient, and (b) for abelian defect groups a block is nilpotent if and only if the inertial quotient is trivial, we have shown that if the inertial quotient of b is isomorphic to that of the given Morita equivalence class representative, then B also has inertial quotient isomorphic to that of the given Morita equivalence class representative.
Putting the reduction techniques of the section together, we have the following that is in a form directly applicable to the proof of Theorem 1.1.Note that we do not need to consider all forms for b here: only those that will arise later.Proposition 5.9.Let G be a finite group and N ✁ G with G/N solvable.Let B be a quasiprimitive 2-block of OG with abelian defect group D of rank at most 4. Suppose that B covers a block b of ON.
for some n and some abelian 2-group P , and Remark 5.10.We have not treated the case that b is Morita equivalent to B 0 (O(A 5 × A 5 ) here since we do not at present know the Picard group of this block.This case will be treated in the main part of the reduction, where we will make use of additional hypotheses.
Finally, the methods above may also be used to prove the following.
for some n 1 , n 2 .By Proposition 5.7(ii) there is only one possibility for the Morita equivalence class of B under the restriction that there is just one simple module.

Proof of the main theorem
We first recall the case where the defect group is normal.Lemma 6.1.Let B be a block of OG for a finite group G with abelian normal defect group D with rank at most 4 and inertial quotient E. Then B is source algebra equivalent to a block of D ⋊ Ê, where E is the inertial quotient of B and Ê/Z( Ê) = E.
Proof.See [42,Theorem 6.14.1].In all cases except E ∼ = C 3 × C 3 , all Sylow subgroups of E are cyclic, so the Schur multiplier is trivial.For C 3 × C 3 the Schur multiplier is 3 and the result follows.
We now state a result used in previous reductions for results concerning Morita equivalence classes of blocks, that encapsulates the use of Fong-Reynolds reductions and [39].It appears in [2] Note that B and C have the same Frobenius categories.
We call the pair (H, C), where C is a block of OH, reduced if it satisfies conditions (R1) and (R2) Lemma 6.2.If the group is clear, then we just say C is reduced.
Before proceeding we recall the definition and some properties of the generalized Fitting subgroup F * (G) of a finite group G. Details may be found in [7].A component of G is a subnormal quasisimple subgroup of G.The components of G commute, and we define the layer E(G) of G to be the normal subgroup of G generated by the components.It is a central product of the components.The Fitting subgroup F (G) is the largest nilpotent normal subgroup of G, and this is the direct product of O r (G) for all primes r dividing |G|.The generalized Fitting subgroup , so in particular G/F * (G) may be viewed as a subgroup of Out(F * (G)).
Proof of Theorem 1.1 Let B be a 2-block of OG for a finite group G with defect group D of rank 4 with [G : O 2 ′ (Z(G))] minimised such that B is not Morita equivalent to any of the blocks listed in the statement of the theorem, or that the inertial quotient of B is not as stated.In the remainder of the proof, the reader may check that the inertial quotients are respected at each step.By Lemma 6.2 we may assume that (G, B) is reduced.
If D ✁ G and B has inertial quotient E B , then by the main result of [36] B is source algebra equivalent to either: (i) D ⋊ E B when E B has cyclic Sylow r-subgroups for every r, that is, for every inertial quotient other than is nilpotent for some j.Let J ⊆ {1, . . ., t} correspond to the orbit of L j under the permutation action of G on the components.Define L J ✁ G to be the product of the L i for i ∈ J, and write b J for the unique block of L J covered by b E .For i ∈ J, since b j is nilpotent, so is b i , hence so is bi .Hence the unique block bJ of L J /Z corresponding to b J is also nilpotent (to see this, observe that as above L J /Z is the quotient of Xi∈J Li by a central 2 ′ -group and that products of nilpotent blocks are nilpotent).It follows that b J is nilpotent by [54] (where the result is stated over k, but Suppose that N ∼ = SL 2 (16) and b is the principal block, which has elementary abelian defect group of order 8 and inertial quotient C 15 .Note that Out(SL 2 (16)) = 1.Hence G = N and we are done in this case.
Suppose that N ∼ = J 1 and b is the principal block, which has elementary abelian defect group of order 8 and inertial quotient C 7 ⋊ C 3 .Note that Out(J 1 ) = 1, so G ∼ = J 1 × C 2 n and we are done in this case.
Suppose that N ∼ = 2 G 2 (3 2m+1 ) for some m ∈ N and b is the principal block, which has elementary abelian defect group of order 8 and inertial quotient C Suppose that N ∼ = Co 3 and b is the non-principal block with elementary abelian defect group of order 8.Note that Out(Co 3 ) = 1, so G ∼ = Co 3 × C 2 n .By [35] b is Morita equivalent to B 0 (O Aut(SL 2 (8))), so B is Morita equivalent to B 0 (O(Aut(SL 2 (8)) × C 2 n )).
We now move on to case (iii) of Proposition 2.7.Then b is Morita equivalent to the principal block of A 5 × P or A 4 × P for some abelian 2-group P of rank at most 2. Since G/N is solvable, Proposition 5.9(iv,v) apply, and B is Morita equivalent to a block on our list, a contradiction.
It remains to consider the case that b is nilpotent covered.By [51, 4.3] b is inertial.Since F * (G) = NZ(G)O 2 (G) and O 2 (G) has rank at most 2, it follows that Out(O 2 (G)) and Out(N), and so Out(F * (G)) are solvable.It follows from Proposition 5.9 that B is Morita equivalent to a block in our list, a contradiction.We have covered each possibility for b as given in Proposition 2.7, so we have established the Morita equivalences in all cases.
Finally we observe that parts (a) and (b)(iv) are proved in [46] and [5].✷ 7 Derived equivalences and Broué's abelian defect group conjecture In this section we prove Corollary 1.3.We first recall Külshammer-Puig classes of blocks.Let D be a defect group for a block B with Frobenius category F .Following the presentation in [42, Section 8.14], a Külshammer-Puig class is an element of H 2 (Aut F (Q)) ∼ = H 2 (E, k × ).As we have used in Lemma 6.1, by [42,Theorem 6.14.1] the Morita equivalence class of a block with normal defect group is determined by the inertial quotient and the Külshammer-Puig class.If E has cyclic Sylow subgroups for all primes, then H 2 (E, k × ) is trivial, so in this case the Morita equivalence class is determined just by E. This is the case for all blocks considered in this paper except for those with inertial quotient C 3 × C 3 .Hence, since inertial quotients are preserved by the Morita equivalences in Theorem 1.1, in order to prove Corollary 1.3 for inertial quotients other than C 3 × C 3 it suffices to show that in each of (b)(i)-(iv) all blocks are derived equivalent when the defect groups are isomorphic (the result is trivial for blocks in Theorem 1.1(a)).This follows from [15,Theorem 4.36] since for the groups in each of (b)(i)-(iv), the principal block of the normalizer of a Sylow 2-subgroup P is Morita equivalent to P ⋊ E, where E is the inertial quotient.This proves Corollary 1.
Let N ✁ G and let b be a block of ON covered by B. We may choose a G-conjugate of D such that D ∩ N is a defect group for b.Write Stab G (b) for the stabiliser of b under conjugation in G.There is a block of OG covering b with a defect group P such that NP/N is a Sylow p-subgroup of Stab G (b)/N.If [G : N] is a power of p, then B is the unique block of OG covering b, and it follows that b is G-stable if and only if G = ND.In this case B and b share a block idempotent.We have the following crucial fact in the case that D is abelian: Proposition 2.1.Let G be a finite group and B a block of OG with abelian defect group D. Suppose that N ✁ G with G = ND and that b is a G-stable block of ON covered by B. Then D acts as inner automorphisms on b.Further, (i) every irreducible character of b is G-stable, hence when further [G : N] = p, extends to p distinct irreducible characters of B, (ii) induction gives a bijection between the projective indecomposable modules for b and those for B. Proof.The assertion that D acts as inner automorphisms follows from [42, Corollary 6.16.3] or [26, Proposition 3.1].The rest is [24, Proposition 2.6(i), (ii)].

Proposition 2 . 4 .
Let B be a block of OG for a finite group G with abelian defect group D. Let (D, B D ) be a B-subpair.Then D = [D, N G (D, B D )] × C D (N G (D, B D )).Suppose there is N ✁ G such that G = DN, then [D, N G (D, B D )] ≤ N. Proof.The factorisation of D follows from [31, Theorem 5.2.3].For the second part, note that [D, N G (D, B D )] = [D, DN N (D, B D )] ≤ N.

Lemma 2 . 6 .
Let B be a block of OG for a finite group G, and let b be a G-stable block of ON for N ✁ G with [G : N] = r a prime different to p. Let E B , E b be the inertial quotient of B, b respectively.Let D be an abelian defect group for B (and for b).Then either E b is isomorphic to a subgroup of E B or E B is isomorphic to a subgroup of E b .Proof.We may take a B-subpair (D, B D ) and a b-subpair (D, b D ) with B D covering b D .If C G (D) ≤ N, then B D = b D and N N (D, b D ) ≤ N G (D, B D ), so the result is immediate in this case.Hence suppose where A is a source algebra for the unique block OG.We have Hom(D/foc(D), O × ) ∼ = C D (N G (D, B D )) = P .The result follows from the description of L(B) in (1) above, as it is clear that the elements of Out D (A) cannot commute with the elements of Out(D, F ) obtained as automorphisms of Q ⋊ C 7 .

Theorem 4 . 2 .
By Lemma 3.6, we have {m 1 , . . ., m s } = {n 1 , . . ., n t } and so P ∼ = Q. 4 Normal subgroups of index 2 Proposition 4.1 (Theorem 3.15 of [25]).Let G be a finite group and N a normal subgroup of G of index p.Now let B be a block of OG with abelian defect group D such that G = ND.Then there exists a block b of ON with the same block idempotent as B and defect group D ∩ N. Moreover there exists a G/N-graded unit a ∈ Z(B), in particular B = p−1 j=0 a j b.Let G, N, B, b and D be as in Proposition 4.1.Suppose further that D ∼ = P ×(C 2 ) 2 , for some finite abelian 2-group P , D∩N ∼ = Q×(C 2 ) 2 , for some subgroup Q ≤ P of index 2, and that b has inertial quotient C 3 and is Morita equivalent to the principal block of O(Q × A 4 ) (respectively O(Q × A 5 )).Then B is Morita equivalent to the principal block of O(P × A 4 ) (respectively O(P × A 5 )).Proof.We follow the proof of [24, Theorem 2.15].Suppose that b is Morita equivalent to the principal block of O(Q×A 4 ) or O(Q×A 5 ) and has inertial quotient C 3 .By Proposition 2.1 D acts as inner automorphisms on b, so every irreducible Brauer character of b is fixed under conjugation in G. Since G/N is a cyclic 2-group, each irreducible Brauer character extends uniquely to G and lies in B (the unique block of G covering b), so l(B) = l(b) = 3.By Proposition 2.5 B and b for A as according to the Morita equivalence class of b.Now I induces an isomorphism of groups Perf(B) ∼ = Perf(B 0 (O(P × A))) via β → I • β • I −1 for β any perfect self-isometry of B, and we denote this isomorphism by I PI .Consider the perfect self-isometry

(
Id Z Irr(P ) , I A ) • I PI (L) • (Id Z Irr(P ) , I A ) −1 : Z Irr(O(P × A 4 )) → Z Irr(O(P × A 4 )) is induced by multiplication by the sign character of an appropriate subgroup.That I PI (L) is of the desired now follows immediately.Composing the perfect isometry induced by the Morita equivalence between b and B 0 (O(Q × A)) with that given by the isomorphism between Q × A and N ′ , we obtain a perfect isometry I Mor : Z Irr(b) → Z Irr(B 0 (ON ′ )).Denote by I N,N ′ the perfect isometry Z Irr(b) → Z Irr(B 0 (ON ′ )) induced by I as in Proposition 3.1.Write I N,N ′ • I −1 Mor = (J ′ , I τ,δ ) in the notation of Theorem 3.5 applied to B 0 (O(R × A)), where J ′ is a perfect self-isometry of OR induced by an O-algebra automorphism α ′ , τ ∈ S 4 and δ ∈ {±1}.By post-composing I with the perfect self-isometry (Id Z Irr(P ) , I τ,δ ) −1 of B 0 (OG ′ ) and post-composing the Morita equivalence b ∼ Mor B 0 (ON ′ ) with that induced by α ′ ⊗ Id B 0 (OA) , we may assume that I N,N ′ = I Mor .Let φ I : Z(B) → Z(B 0 (OG ′ )) be the isomorphism of centres induced by I as in Lemma 3.1 and let M be the B 0 (ON ′ )-b-bimodule inducing the above Morita equivalence b ∼ Mor B 0 (ON ′ ).Since I N,N ′ = I Mor , by Proposition 3.1 we have that φ I | Z(b) = φ I N,N ′ : Z(b) → Z(B 0 (ON ′ )) is the isomorphism of centres induced by the Morita equivalence.In other words, φ I (b)m = mb, for all b ∈ b, m ∈ M.

Theorem 5 . 1 (
[56]).With the notation above, suppose further that [G : N] is a power of p, so thatG = DN.Let E = N G (D, B D )/C G (D)suppose that E is cyclic and acts freely on [N G (D, B D ), D] \ {1} (that is, all orbits have length |E|).Suppose that b is inertial, i.e., there is a basic Morita equivalence with O(Q ⋊ E).Then B is Morita equivalent to O(D ⋊ E).

Lemma 5 . 4 .
With the notation above, suppose G/N has prime order r different to p and that Out(f bf ) has cyclic Sylow r-subgroups of order r.Then there are precisely two possibilities for Morita equivalence class of B, one of which is that B is source algebra equivalent to b. Proof.The trivial homomorphism G/N → Out(f bf ) corresponds to the case that B is source algebra equivalent to b by Proposition 2.3, since G = G[b].Consider nontrivial α : G/N → Out(f bf ), and consider a block B such that f Bf is a crossed product of f bf with G/N corresponding to α.Then for each ϕ ∈ Aut(G/N) we have that α • ϕ also realises f Bf as a crossed product.This accounts for all possible homomorphisms α : G/N → Out(f bf ).

Proposition 5 . 7 .
Let G be a finite group and N ✁ G with [G : N] = r, where r is an odd prime.Let B be a 2-block of OG covering a G-stable block of b of ON. (i) Suppose that b is Morita equivalent to O(((C 2 n ) 3 ⋊ C 7 ) × P ) where n ∈ N and P is a cyclic 2-group.Suppose r = 3.Then B is either source algebra equivalent to b or Morita equivalent to O(((C 2 n ) 3 ⋊ (C 7 ⋊ C 3 )) × P ).Further, if b is known to have inertial quotient C 7 , then in the latter case B has inertial quotient C 7 ⋊ C 3 .Suppose r = 7.Then B is either source algebra Morita equivalent to b or is nilpotent.If r is an odd prime other than 3 and 7, then B is source algebra equivalent to b. (ii) Suppose that b is Morita equivalent to O(G m × P ), where m ∈ N and P is an abelian 2-group with rank at most 2. Let D be the defect group of b.If r = 3, then B is source algebra equivalent to b.If r = 3 and P ∼ = (C 2 n ) for some n ∈ N, then B is either nilpotent, source algebra equivalent to b, or Morita equivalent to

7 and that r = 3 .
It follows from Lemma 2.6 that B has inertial quotient C 7 or C 7 ⋊ C 3 .By Remark 5.6 we observe that we may consider elements of T (b).By the description of T (b) in Section 2.5 we know that T (b) maps to a subgroup of Out(D, F ) ∼ = C 3 .If this subgroup is trivial, then we may not construct any nontrivial homomorphism G/N → T (b) and we are done.Hence suppose the subgroup has order three.Then the action of G induces an element of order three in Aut(D, F ), so that B must have inertial quotient C 7 ⋊ C 3 .
then B is inertial with inertial quotient 1, (C 3 ) 2 , C 5 or C 15 .(iv) If b is Morita equivalent to a block with normal defect group and has inertial quotient (C 3 ) 1 , then B is Morita equivalent to one of: (a) OD; (b) O(G n × P ), where D ∼ = (C 2 n ) 2 × P , and B has inertial quotient (C 3 ) 1 ; (c) O(G n 1 × G n 2 ), where D ∼ = (C 2 n 1 ) 2 × (C 2 n 2 ) 2 , and B has inertial quotient C 3 × C 3 ; (d) a non-principal block of O(D ⋊ 3 1+2 + ) and B has inertial quotient C 3 × C 3 ; (v) If b is Morita equivalent to B 0 (O(A 5 × Q)) for some Q ≤ D and has inertial quotient of type (C 3 ) 1 , then B is Morita equivalent to one of: (a) B 0 (O(A 5 × P )), where D ∼ = (C 2 ) 2 × P , and B has inertial quotient(C 3 ) 1 ; (b) B 0 (O(A 5 × G n )), where D ∼ = (C 2 ) 2 × (C 2 n ) 2, and B has inertial quotientC 3 × C 3 .(vi) If b is Morita equivalent to B 0 (O(A 5 × G n )) for some n ≥ 1, then B is Morita equivalent to b or B 0 (O(A 5 × (C 2 n ) 2 )), where D ∼ = (C 2 ) 2 × (C 2 n ) 2 ,and B has inertial quotient C 3 × C 3 or (C 3 ) 1 respectively.Proof.Since b is G-stable and G/N is solvable, it follows from Lemma 2.2 that DN/N is an abelian Sylow 2-subgroup of G/N, which must then have 2-length at most one.By Proposition 2.1 D ≤ G[b], which must also have 2-length at most one.Hence there are normal subgroupsN i of G such that N = N 0 ✁ N 1 ✁ N 2 ✁ N 3 = G[b] with N 1 /N 0 , N3/N 2 of odd order and N 2 /N 1 a 2-group, so in particular N 2 = N 1 D. Write b i for the unique block of ON i covered by B. Let (D, (b 2 ) D ) be a b 2 -subpair.By Proposition 2.4 D ∼ = [D, N N 2 (D, (b 2 ) D )] × C D (N N 2 (D, (b 2 ) D )) with [D, N N 2 (D, (b 2 ) D )] ≤ N 1 (and so [D, N N 2 (D, (b 2 ) D )] ≤ N).By Proposition 2.3 b is source algebra equivalent to b 1 , with the same inertial quotient as b.If furthermore b is inertial, then b 1 is also inertial.By Proposition 2.5 b 2 has inertial quotient isomorphic to that of b, with the same action (this last statement uses the fact that in this case there is a unique action given the isomorphism type of the inertial quotient and the order of commutator part[D, N N 2 (D, (b 2 ) D )] of D).(i) Suppose that b, and so b 2 , has inertial quotient C 7 and that b is inertial.By Theorem 5.1 b 2 is Morita equivalent (but not necessarily basic Morita equivalent) to O(D ⋊ C 7 ).Note that C D (N N 2 (D, (b 2 ) D )) is cyclic.By Proposition 2.9 Pic(b 2 ) ∼ = (C 7 ⋊C 3 )×(C D (N N 2 (D, (b 2 ) D ))⋊Aut(C D (N N 2 (D, (b 2 ) D ))).Now consider G[b 2 ].Again applying Proposition 2.3, b 2 is source algebra equivalent to the unique block c of G[b 2 ] covered by B. Since G/G[b 2 ] is an odd order subgroup of Pic(b 2 ), it is isomorphic to a subgroup of C 7 ⋊ C 3 .Suppose that G/G[b 2 ] ∼ = C7 .By Proposition 5.7 either B is nilpotent or it is Morita equivalent to c with inertial quotient C 7 .If G/G[b 2 ] ∼ = C 3 , then again by Proposition 5.7 B is Morita equivalent to c or O(D ⋊ (C 7 ⋊ C 3 )), with the appropriate inertial quotient.Suppose G/G[b 2 ] ∼ = C 7 ⋊ C 3 .Write G 1 for the preimage of O 7 (G/G[b 2 ]) in G and B 1 for the unique block of G 1 covered by B. Then by Proposition 5.7 B 1 is either nilpotent or Morita equivalent to c. Applying Proposition 5.7 again, in the first case B is either nilpotent or Morita equivalent to a block with normal defect group and inertial quotient (C 3 ) 1 .In the second, B is either Morita equivalent to c or to O(D ⋊(C 7 ⋊C 3 )).In each of these cases by Proposition 5.7 the inertial quotient is as stated.(ii) and (iii) Suppose that b, and so b 2 , has inertial quotient (C 3 ) 2 , C 5 , C 15 or C 3 ×C 3 , and that b is inertial.We have C D (N N 2 (D, (b 2 ) D )) = 1, so D = [D, N N 2 (D, (b 2 ) D )] ≤ N 1 , i.e., N 1 = N 2 and G/N has odd order.The result then follows from Theorem 5.2.(iv) Suppose that b, and so b 1 , b 2 , has inertial quotient (C 3 ) 1 .Hence b 1 is Morita equivalent to O(G n × Q) for some n and Q ≤ D. Recalling that [D, N N 2 (D, (b 2 ) D )] ≤ N 1 , by Theorem 4.2 b 2 is Morita equivalent to O(G n × Q), where D ∼ = (C 2 n ) 2 × P .Applying Proposition 2.3 b 2 is source algebra equivalent to the unique block c of G[b 2 ] covered by B. By Proposition 2.9 Pic(b 2 ) ∼ = S 3 × (P ⋊ Aut(P )).Suppose that P is not homocyclic.Since G/G[b 2 ] is an odd order subgroup of Pic(b 2 ) and | Pic(b 2 )| 2 ′ = 3, it is isomorphic to a subgroup of C 3 .By Proposition 5.7 B is either Morita equivalent to c or is nilpotent.Suppose that P is homocyclic.Then Pic(b 2 ) ∼ = S 3 × (P ⋊ S 3 ) and G/G[b 2 ] is isomorphic to a subgroup of C 3 × C 3 .Let H be the kernel of the action of G on the irreducible Brauer characters of c and let B H be the unique block of OH covered by B. By Proposition 2.9 [G : H] divides 3. Suppose that [G : G[b 2 ]] = 3.By Proposition 5.7 the possible Morita equivalence classes for B are represented by the four blocks listed.Suppose that [G : G[b 2 ]] = 9.If G = G[c], then by Proposition 2.3 B is Morita equivalent to c and we are done.If [G : G[c]] = 3, then B 1 is Morita equivalent to c, and again there are four possibilities for the Morita equivalence class of B and these are as listed.Hence suppose that G[c] = G[b 2 ], so that G[b 2 ] < H < G.The subgroup of P ic(c) of self-equivalences that preserve all irreducible Brauer characters is isomorphic to S 3 .Hence there is one possibility for the Morita equivalence class for B H (as we are excluding the case that H acts as inner automorphisms), that B 1 is Morita equivalent to O(G n 1 × G n 2 ).One may check as in (i) that the inertial quotients are as stated.(v) Suppose that b, and so b 1 , b 2 , is Morita equivalent to B 0 (O(A 5 × Q)) for some Q ≤ D. By Theorem 4.2 b 2 is Morita equivalent to B 0 (O(A 5 × P )), where D ∼ = (C 2 ) 2 × P .By Proposition 2.3 b 2 is source algebra equivalent to the unique block of G[b 2 ] covered by B. By Proposition 2.9 Pic(b 2 ) ∼ = C 2 × (P ⋊ Aut(P )).Since G/G[b 2 ] is an odd order subgroup of Pic(b 2 ) and P has rank at most 2, it is isomorphic to a subgroup of C 3 .If P is not homocyclic, then G = G[b 2 ] and we are done.Suppose P is homocyclic, i.e., P ∼ = (C 2 n ) 2 for some n.We may suppose [G : G[b 2 ]] = 3.The result then follows by Proposition 5.7.(vi) Suppose that b is Morita equivalent to B 0 (O(A 5 × G n )) for some n ≥ 1.As in (ii) G/N has odd order, so N 1 = N 2 and b is source algebra equivalent to b 3 .By Proposition 2.9 Pic(b) ∼ = C 2 × S 3 .Since G/G[b] is an odd order subgroup of Pic(b), it is isomorphic to a subgroup of C 3 .If G[b] = G, then b is source algebra equivalent to B and we are done.If [G : G[b]] = 3, then we are done by Proposition 5.7.

Proposition 5 . 11 .
Let D ∼ = (C 2 n 1 ) 2 × (C 2 n 2 ) 2 for some n 1 , n 2 ∈ N and consider G = D ⋊ 3 1+2 + , where the centre of 3 1+2 + acts trivially.The 2-blocks of OG correspond to the simple modules of Z(3 1+2 + ), and the two non-principal blocks are Morita equivalent.Further, these blocks are Morita equivalent to the two non-principal blocks of O(D ⋊ 3 1+2 − ).Proof.Let B be any faithful 2-block of G = D ⋊3 1+2 + or D ⋊3 1+2 − .Then l(B) = 1.Take a maximal subgroup N of G and a block b of N covered by B. Then N ∼ = (D ⋊C 3 )×C 3 or D ⋊ C 9 and without loss of generality b is Morita equivalent to where Z(3 1+2 ) acts trivially on D. By Proposition 5.11 the non-principal blocks of D ⋊ 3 1+2 + and D ⋊ 3 1+2 − form a single Morita equivalence class.Hence B is Morita equivalent to a block listed in the statement of the theorem, a contradiction.Hence D is not normal in G. Let b * be the unique block of OF * (G) covered by B.Write L 1 , . . ., L t for the components of G, soE(G) = L 1 • • • L t ✁ G.Note that G permutes the L i .There must be at least one component, since otherwise b * is nilpotent and soF * (G) = Z(G)O 2 (G).But O 2 (G) ≤ D and D is abelian, so we would have D ≤ C G (F * (G)) ≤ F * (G) = Z(G)O 2 (G), so that D ✁ G, a contradiction.Write b E for the unique block of E(G) covered by B and b i for the unique block of L i covered by b E .We claim that no b i can be nilpotent (in our minimal counterexample).Let Z = O 2 (Z(E(G))).Write bE for the unique block of E(G) := E(G)/Z corresponding to b E and bi for the unique block of Li corresponding to b i .WritingM := L 1 × • • • × L t , where L i := L i Z/Z, there is a 2 ′ -group W ≤ Z(M)and a block b M of M with W in its kernel such that E(G) = M/W and b M is isomorphic to bE .Then D ∩ E(G) is a defect group for b E , (D ∩ E(G))/Z is a defect group for bE and b M has defect groups isomorphic to (D ∩ E(G))/Z.Then bi has defect group D i = ((D ∩ E(G))Z) ∩ Li .We have that b M = b1 ⊗ • • • ⊗ bt and b M has defect group D 1 × • • • × D t .Suppose b j
3 when E ∼ = C 3 × C 3 .Now suppose that B has inertial quotient E ∼ = C 3 ×C 3 .We must determine whether the Brauer correspondent b of B in N G (D) is Morita equivalent to O(D ⋊ E) or to O(D ⋊ 3 1+2 + ).A nonprincipal block of D ⋊ 3 1+2 + has just one simple module whilst all other blocks in Theorem 1.1(b)(v) have nine simple modules.However by [53] and [33, Proposition 5.5] l(B) = l(b), so we may distinguish the Morita equivalence class of b by l(B).By [15, Theorem 4.36] the principal blocks occurring in (b)(v) are derived equivalent, so Corollary 1.3 follows.
Proposition 2.3.Let G be a finite group and B a block of OG with defect group D. Let N ✁ G with D ≤ N and suppose that B covers a G-stable block b of ON.
Let B be a block of OG for a finite group G with abelian defect group D and let (D, B D ) be a B-subpair.Let N ✁G.Suppose that B covers a G-stable block b of ON, and that [24,roceed as in the proof of[24, Theorem 2.11].The set of projective indecomposable characters (characters of projective indecomposable modules) is prj(O(P × A 4 )) = {χ P 1 , χ P 2 , χ P 3 }, where χ P 24, Proposition 2.8] together with the observation that by[13, A1.3] OA 4 and B 0 (OA 5 ) are perfectly isometric.Theorem 3.5.Let P be a finite abelian 2-group.Every perfect self-isometry of O(P × A 4 ) is of the form (J, I σ,ǫ ), where J is a perfect isometry of OP induced by an O-algebra automorphism, σ ∈ S 4 and ǫ ∈ {±1}.Proof.j := θ∈Irr(P ) in full, but was extracted from the first part of the proof of [21, Proposition 4.3].Lemma 6.2.Let G be a finite group and B a block of OG with defect group D. Then there is a finite group H and a block C of OH such that B is basic Morita equivalent