Picard groups of blocks with normal defect groups ∗

Let b be a block with normal abelian defect group and abelian inertial quotient. We prove that every Morita auto-equivalence of b has linear source. We note that this improves upon results of Zhou and Boltje, Kessar and Linckelmann. We also prove that Picent(b) is trivial which is conjectured to be the case for all blocks.


Introduction
Let p be a prime, (K, O, k) a p-modular system with k algebraically closed and b a block of OH, for a finite group H. We always assume that K contains all |H| th roots of unity. The Picard group Pic There are many open problems concerning Picard groups. It is proved in [7,Corollary 1.2] that Pic(b) is finite. However, it is yet to be proved that Pic(b) is always bounded in terms of a function of the defect group. There are also no known examples of a block b with Pic(b) = E(b). Our main result (see Theorem 6.3) is as follows: Theorem. Let b be a block with a normal abelian defect group and abelian inertial quotient, then Pic(b) = L(b).
We note that this improves upon a result of Zhou [16,Theorem 14]. Zhou proves that if b = O(D ⋊ E), where D is an abelian p-group and E is an abelian p ′ -group of Aut(D), then Pic(b) = L(b). We can also compare with a result of Boltje, Linckelmann and Kessar [3,Proposition 4.3], where it is assumed in addition that [D, E] = D but the result is that Pic(b) = T (b). Note that this result follows immediately from Corollary 6.4.
The following notation will hold throughout this article. We define : O → k to be the natural quotient map and, for a finite group H, we extend this to the corresponding ring homomorphism : OH → kH. A block b of H will always mean a block of OH. We set Irr(H) (respectively IBr(H)) to be the set of ordinary irreducible (respectively irreducible Brauer) characters of H and Irr(b) ⊆ Irr(H) (respectively IBr(b) ⊆ IBr(H)) the set of ordinary irreducible (respectively irreducible Brauer) characters lying in the block b. If N ✁ H and χ ∈ Irr(N ), then we denote by Irr(H, χ) the set of irreducible characters of H appearing as constituents of χ ↑ H . Similarly, we define Irr(b, χ) := Irr(b) ∩ Irr(H, χ). 1 H ∈ Irr(H) will designate the trivial character of H. We use e b ∈ OH to denote the block idempotent of b and e χ ∈ KH to denote the character idempotent associated to χ ∈ Irr(H). Finally, we set The article is organised as follows. In §2 we establish some preliminaries about abelian p ′ -groups acting on abelian p-groups. We introduce a particular block with normal abelian defect group and abelian inertial quotient in §3. §4 is concerned with perfect isometries and how they relate to our main theorem. In §5 we study the specifc case of a block with one simple module in greater detail and our main theorem is proved in §6.
2 Abelian p ′ -groups acting on abelian p-groups Definition 2.1. Let H be a finite abelian p ′ -subgroup of Aut(P ), for some abelian p-group P . We say H acts on P . If there exists a non-trivial direct decomposition P ∼ = P 1 × P 2 such that P 1 and P 2 are both H-invariant, then we say H acts decomposably on P . Otherwise we say H acts indecomposably on P .
We note that our definition of H acting on P is non-standard in that we are demanding that H is a finite p ′ -group, P is a finite abelian p-group and that H is a subgroup of Aut(P ) (usually called a faithful action) and not just that we have a group homomorphism H → Aut(P ). Whenever we have H acting on P we always have the semi-direct product P ⋊ H, defined through this action, in mind. We will borrow notation from this setup, for example C P (H) will denote the set of fixed points in P under the action of H. Proof. By Lemma 2.3, P = [P, H] × C P (H). Therefore, if P has a non-trivial fixed point we can certainly construct some non-trivial fixed point of Irr(P ). The converse follows since we can identify the action of H on P with that of H on Irr(Irr(P )).
We denote by Φ(P ) the Frattini subgroup of P .
Lemma 2.5. Let H act indecomposably on P ∼ = (C p n ) m , for some m, n ∈ N. Then we have an induced action of H on P/Φ(P ) ∼ = (C p ) m and this action is also indecomposable.
Proof. The fact that we have an induced action follows from [9, §5, Theorem 1.4]. Assume H acts decomposably on P/Φ(P ). Let x ∈ P \Φ(P ) be such that xΦ(P ) is contained in a non-trivial H-invariant direct factor of P/Φ(P ) and consider the smallest H-invariant subgroup Q of P containing x. Certainly {1} < Q < P and Q Φ(P ). So, by Remark 2.2, there exists some H-invariant homocyclic direct factor Q ′ of Q also satisfying {1} < Q ′ < P and Q ′ Φ(P ). So Q ′ ∼ = C m ′ p n for some 1 ≤ m ′ < n. In particular Q ′ is an H-invariant direct factor of P . Again by Remark 2.2, this contradicts the indecomposablity of the action of H on P .
In what follows, J(kP ) will denote the Jacobson radical of kP . Lemma 2.6. Let H act indecomposably on P ∼ = (C p n ) m , for some m, n ∈ N.
1. H is cyclic and if g is a generator of H then g has m distinct eigenvalues {λ = λ p m , λ p , . . . , λ p m−1 }, as a linear transformation of k ⊗ Fp P/Φ(P ).
Therefore, since by Lemma 2.5 we have an indecomposable action of H on P/Φ(P ), we assume for the remainder of the proof that P is elementary abelian.
1. We identify P with F m p and view H as a subgroup of G := GL m (F p ). Let g be an element of maximal order in H. We factorise the characteristic polynomial of g into irreducible factors where f i (X) and f j (X) are coprime for i = j. We first note that is a non-trivial H-invariant subspace of F m p . Therefore, since H acts indecomposably, we must have f 1 (g) = 0, in particular s = 1 and f 1 (X) has degree d := m/n 1 . It follows that o(g)|(p d − 1) and d is the smallest positive integer satisfying this condition, where o(g) is the order of g. Then C G (g) ∼ = GL n1 (F p d ) and g is represented in C G (g) by the scalar matrix with λ's on the diagonal for some λ ∈ F p d a root of f 1 (X) (see for example [8,Proposition 1A]).
Certainly o(h)|o(g) for each h ∈ H ≤ C G (g) and so the characteristic polynomial of h in C G (g) ∼ = GL n1 (F p d ) must factorise into linear factors. Exactly as for g in GL m (F p ), the characteristic polynomial of h in GL n1 (F p d , must be the power of an irreducible polynomial. Therefore, h is also a scalar matrix in C G (g) and, since o(h)|o(g), it must be a power of g proving H is cyclic. In particular, F m p decomposes into the direct sum of n 1 H-invariant subspaces and so, in fact, n 1 = 1. The eigenvalues of g are now just the roots of f 1 (X). Since f 1 (X) has degree m, this proves the first part of the lemma.
for some a i ∈ OP . Then By calculating in F p P we have that The first claim follows.
If p = 2 and n = 1, then ( 2 (P ) and the second claim follows from the comments preceding the Lemma.

O(D ⋊ E)e ϕ and its characters
We set the following notation that will hold for the rest of the article. Let D be a finite abelian p-group, E a finite p ′ -group and Z ≤ E a central, cyclic subgroup such that L := E/Z is abelian. Let L act on D set G := D ⋊ E through this action. We study B := OGe ϕ , where ϕ is a faithful character of Z. Since D ✁ G, any block idempotent of OG is supported on C G (D) = D × Z. Therefore, B ϕ is a block of OG with defect group D. Set Before we go on to describe the irreducible characters of B we focus on Irr(E, ϕ). 1. If χ 1 , χ 2 ∈ Irr(E, ϕ), then there exists θ ∈ Irr(E, 1 Z ) such that χ 1 .θ = χ 2 .

Proof.
1. Let χ ∈ Irr(E, ϕ). Then Since every element of Irr(E, ϕ) appears as a constituent of ϕ ↑ E and 1 Z ↑ E only has constituents in Irr(E, 1 Z ), the claim follows.
2. Since Z(E) is abelian, the first statement is clear. Now for all g ∈ E h∈E/CE(g) Since ϕ is faithful, this is zero unless g ∈ Z(E). In other words, Z(KEe ϕ ) = KZ(E)e ϕ . So the e ψ 's, as ψ ranges over Irr(Z(E), ϕ), are all the character idempotents of KEe ϕ . Setting χ ψ ∈ Irr(E, ϕ) to be such that e χ ψ = e ψ , the claim follows by considering the left KEe ϕ -module isomorphism for all ψ ∈ Irr(Z(E), ϕ).
We now describe Irr(B). Let λ ∈ Irr(D) and set E λ ≤ E to be the stabiliser of λ in E. Choose χ ∈ Irr(E λ ) and define (λ, χ) ∈ Irr(D ⋊ E λ ) by for g ∈ D and h ∈ E λ . Note that ker(λ)⋊E λ is a normal subgroup of D⋊E λ and so we can uniquely extend λ to a character of D ⋊ E λ with kernel ker(λ) ⋊ E λ . (λ, χ) is just this extension tensored with the inflation of χ to D ⋊ E λ . 1. The irreducible characters of B are precisely of the form (λ, χ) ↑ G for some λ ∈ Irr(D) and χ ∈ Irr(E λ , ϕ).

There is a bijection
, for all g ∈ G p ′ and Inf G E denotes the inflation of a character from E ∼ = G/(D × Z) to G.

Through this bijection we can identify the decomposition map
Proof.
1. Every simple kG-module must have D in its kernel and the decomposition map is a bijection on p ′ -groups, so we can associate a unique irreducible character of E to each irreducible Brauer character of G. The first part then follows from the fact that an irreducible Brauer character ψ of G is in B if and only if its restriction to Z is ϕ ⊕ψ(1) .
2. Every χ ∈ Irr(B) restricted to Z is ϕ ⊕χ(1) and so the restriction map is well-defined. The claim now follows by noting that irreducible Brauer characters of B are completely determined by their restriction to E.
Proof. The existence of such a σ Br is just the statement that any Morita autoequivalence permutes IBr(B), which we identify with Irr(E, ϕ) via Lemma 3.3. The uniqueness follows from the fact that every element of Irr(E, ϕ) inflates to an element of Irr(B).

By Lemma 2.3 we may decompose
Lemma 3.5. The subset of irreducible characters of B that reduce to some number of copies of the same irreducible Brauer character is Part (2) of Lemma 3.3 gives that (λ, χ) ↑ G reduces to some number of copies of the same irreducible Brauer character if and only if (λ, χ) ↑ G ↓ E = χ ↑ E is the sum of some number of the same irreducible character of E. By part (2) of Lemma 3.1, this happens if and only if χ ↑ E ↓ Z(E) is the sum of some number of the same irreducible character of Z(E). It follows from the Mackey Before proceeding we note that if ω ∈ C is a primitive (p n ) th -root of unity, for some n ∈ N, then In particular,   (a) There exists an O-free OG-module V affording (λ, χ) with x acting as the identity on Proof.
1. If such a V exists then certainly λ(x) ≡ 1 mod p for all x ∈ D 1 . However, 1 − ω ∈ pO for some p th -power root of unity ω ∈ O if and only if ω = 1.
. Certainly x acts as the identity on O p ⊗ O U , for all x ∈ D 1 and therefore setting V := U ↑ G proves the claim.

Perfect isometries
Let H be a finite group and b a block of OH. We write prj(b) for the set of characters of projective indecomposable b-modules.
Let H ′ be another finite group and b ′ a block of OH ′ . A perfect isometry between b and b ′ is an isometry such that (Note that by an isometry we mean an isometry with respect to the usual inner products on Z Irr(b) and Z Irr(b ′ ). In particular, for all χ ∈ Irr(b), I(χ) = ±χ ′ for some χ ′ ∈ Irr(b ′ )).
Remark 4.2. An alternative way of phrasing the condition that I K induces an isomorphism between We need a small lemma before continuing.
Let H be a finite group, N a normal subgroup and χ ∈ Z Irr(N ).
Proof. χ H is zero outside of N so we need only prove that This is now clear, as χ ↑ H ↓ N is just the sum of the Hconjugates of χ, each appearing with multiplicity [Stab H (χ) : N ].
The following lemma is the main result of this section.
Proof. Let's fix some ξ ∈ Irr(E, ϕ) and define θ ∈ Irr(D 2 ) by Let D ′ ✁ D 1 ⋊ E be properly contained in D 1 and maximal with respect to these two conditions. Define E ′ ≤ E to be the subgroup inducing the identity on D 1 /D ′ . In particular, D 1 /D ′ is elementary abelian and E/E ′ acts indecomposably on D 1 /D ′ . Therefore, by part (1) of Lemma 2.6, E/E ′ is cyclic and, by part (2) of the same Lemma and Lemma 2.4, E ′ is the stabiliser in E of any non-trivial character of D 1 /D ′ inflated to D 1 .
In the final part of the proof we prove that the intersection of all possible choices for E ′ is Z. We will have then proved that the set of χ ∈ Irr(E, ϕ) that satisfy (1) is closed under tensoring with elements of Irr(E, 1 Z ). By part (1) of Lemma 3.1 we will then be done.
Let's decompose where E/C E (Q i ) acts indecomposably on Q i . Now Lemma 2.5 implies that E/C E (Q i ) also acts indecomposably on each Q i /Φ(Q i ). In particular, for each is a valid choice for D ′ and C E (Q i ) a valid choice for E ′ . Finally, by the definition of Z, and the proof is complete.

Blocks with one simple module
Throughout this section we assume that B has, up to isomorphism, a unique simple module. By part (1) of Lemma 3.3 and part (2) of Lemma 3.1, this implies that Z = Z(E). For each g ∈ L = E/Z we define whereg andh represent lifts to E of g and h respectively. Note it is easy to check that φ g is a well-defined group homomorphism.
is an isomorphism of groups.
Proof. This is just [10, Lemma 4.1] and its proof.
We now introduce some further notation. First decompose where E/C E (P i ) acts on each P i indecomposably and P i ∼ = (C p n i ) mi . We choose this decomposition such that D 1 = P 1 × · · · × P t and D 2 = P t+1 × · · · × P n , for some 1 ≤ t ≤ n. In particular, m i = 1 for all i > t. We now state and prove a partial analogue of [ where d is the dimension of the unique simple Bmodule. In particular, A is basic.
2. There exist X ij ∈ A, for 1 ≤ i ≤ n and 1 ≤ j ≤ m i that generate A as an O-algebra. Furthermore, 3. There exist p ′ -roots of unity q i1j1,i2j2 ∈ O × such that (e) Let 1 ≤ i 1 ≤ t and 1 ≤ j 1 ≤ m i1 . For all 1 ≤ j 2 ≤ m i1 with j 2 = j 1 , there exist 1 ≤ i ≤ t and 1 ≤ j ≤ m i such that q i1j1,ij = q i1j2,ij .

We can choose the
we have the following identification of ideals 5. For all i > t, Moreover, if p = 2 and D 1 is elementary abelian, then for each 1 ≤ i ≤ t and 1 ≤ j ≤ m i , m i > 1 and Proof. (1)  We, therefore, define A := C B (OEe ϕ ). (1) and (3) of Lemma 2.6, for each 1 ≤ i ≤ n, J(kP i )/J 2 (kP i ) decomposes into m i non-isomorphic linear representations of E,

By parts
with respect to the conjugation action of E on P i . As p ∤ |E|, we can decompose Now set X ij = h ij W ij e ϕ , for all 1 ≤ i ≤ n and 1 ≤ j ≤ m i . We first note that for all h ∈ E and so X ij ∈ C B (OEe ϕ ). Note that the X ij = h ij w ij e ϕ ∈ k ⊗ O A are precisely the X i 's constructed in the proof of [10,Corollary 4.3]. In particular, k ⊗ O B forms a basis for C kB (kEe ϕ ). So B is an O-linearly independent set and B O is an O-summand of B. Therefore, since B O ⊆ C B (OEe ϕ ) and 3. By Lemma 2.7, W p n i ij ∈ pJ O (P i ) for all i and j. Therefore, since

Parts (a) and (b) follow immediately from this definition. Part (c) holds since
As representations of L, we claim that the For part (e) we suppose the contrary, that is there exists 1 ≤ j 2 ≤ m i1 with j 1 = j 2 such that q i1j1,ij = q i1j2,ij for all 1 ≤ i ≤ t and 1 ≤ j ≤ m i . In other words, , for all such i and j. Since contradicting part (1) of Lemma 2.6.
4. First note that the w ij 's, for 1 ≤ i ≤ t and 1 ≤ j ≤ m i , form a basis for Therefore, as required.

5.
Similarly to the proof of part (4), forms an O-span of OP i , for all 1 ≤ i ≤ n. (Note that we need not take higher powers of W ij since, as noted in the proof of part (3), W p n i ij ∈ pJ O (OP i ).) By comparing O-ranks, (6) must actually form a basis of OP i . The first statement now follows from the fact that, since E commutes with From now on we assume p = 2 and D 1 is elementary abelian. We fix some 1 ≤ i ≤ t. If m i = 1, then P i has no non-trivial automorphisms and so P i ≤ C D (E) = D 2 , a contradiction. So we must have m i > 1. Let 1 ≤ j ≤ m i , where we are considering j modulo m i . Of course, n i = 1 and so, by Lemma 2.7, W 2 ij = 2y for some y ∈ J O (P i )\J O (P i ) 2 . Therefore, y O affords ̺ 2 j i and so, by the construction of the W il 's and using the basis from (6), y = λ j W i(j+1) + W , for some λ j ∈ O and Since y / ∈ J O (P i ) 2 , in fact λ j ∈ O × . By the construction of the h il 's, we also have h 2 ij Z = h i(j+1) Z and so h 2 ij e ϕ = µ j h i(j+1) e ϕ , for some µ j ∈ O × . Therefore, for all mi l=1 W ǫ il il appearing with non-zero coefficient in W . We have now shown that for all 1 ≤ j ≤ m i . Since raising to the power 2 mi − 1 on O × is a surjective map, there exists α 1 ∈ O × such that Now set α j+1 = α 2 j λ j µ j , for all 1 ≤ j ≤ m i . Note that by this definition In other words, α j is well-defined when we consider j mod m i . Therefore, We may express each element of A k uniquely as a k-linear combination of elements of k ⊗ O B and in the following lemma we refer to the terms of this k-linear combination. Similarly we refer to the terms of an element of A p and A I . Set We denote by X a , the monomial X at+1 t+1 . . . X an n ∈ k[X i1 ] t+1≤i≤n , where a = (a t+1 , . . . , a n ) ∈ A. For a, b ∈ A, a + b signifies the componentwise sum, when this is still in A. We have a partial order on A given by a ≤ c if and only if there exists b ∈ A such that a + b = c. In this case note that X a+b = X a X b . We adopt the same notation for monomials in In what follows, when it is clear from the context, we will denote by X ij its image in A k , A p or A I , for 1 ≤ i ≤ n and 1 ≤ j ≤ m i . The next lemma should be thought of as an attempt to generalise [2, Lemma 2.3], where automorphisms of A k are studied. In [2] the term special generating set is used to describe certain subsets of A k . We note that the image of the X ij 's under a k-algebra automorphism of A k is a special generating set. We use this fact below.
Then there exists some 1 ≤ r 0 ≤ n and 1 ≤ s 0 ≤ m r0 such that X r0s0 appears with unit coefficient in φ(X rs ). If, in addition, 1 ≤ u ≤ n and 1 ≤ v ≤ m u such that X u0v0 appears with unit coefficient in φ(X uv ), for some 1 ≤ u 0 ≤ n and 1 ≤ v 0 ≤ m u0 , then q rs,uv = q r0s0,u0v0 . In particular, there does not exist 1 ≤ s ′ 0 ≤ m r0 different from s 0 such that X r0s ′ 0 also appears with unit coefficient in φ(X rs ). We have all the analogous results for A I .

All
3. Assume p = 2 and D 1 is elementary abelian. All O I -algebra automorphisms of A I leave T I invariant. Proof.
1. We prove all the results for A p and A I simultaneously.
As noted in the proof of Lemma 5.2, the X ij ∈ A k are precisely the X i 's constructed in the proof of [10,Corollary 4.3]. Certainly A k is local, it is the basic algebra of a block with one simple module, and so J(A k ) is the ideal generated by the X ij 's. In particular, the X ij 's form a basis of J(A k )/J 2 (A k ). Now φ induces a k-algebra automorphism φ k of A k . Therefore, there exists some X r0s0 appearing with non-zero coefficient in φ k (X rs ). Since an element x ∈ O is invertible if and only if x / ∈ J , the first claim follows.
Suppose X r0s0 ∈ A k (respectively X u0v0 ∈ A k ) appears with non-zero coefficient in φ k (X rs ) (respectively φ k (X uv )). We note that, since p ′ -roots of unity in k lift uniquely to O, we need only prove that q rs,uv = q r0s0,u0v0 . By [2, Lemma 2.3(i)], for any X ij that appears with non-zero coefficient in φ k (X rs ), q ij,u0v0 = q r0s0,u0v0 and an analogous statement for any X ij appearing with non-zero coefficient in φ k (X uv ). In particular, if (r, s) = (u, v), then q r0s0,u0v0 = q u0v0,u0v0 = 1 = q rs,uv .
Therefore, we assume (r, s) = (u, v). In this case X rs X uv ∈ J 2 (A k )\J 3 (A k ) and so there must exist some X r ′ appears with non-zero coefficients λ rs λ uv in φ k (X rs X uv ) and λ rs λ uv q −1 For the final claim suppose such an s ′ 0 does exist and choose 1 ≤ u 0 ≤ t and 1 ≤ v 0 ≤ m u0 such that q r0s0,u0v0 = q r0s ′ 0 ,u0v0 . The existence of u 0 and v 0 is guaranteed by part (3e) of Lemma 5.2. Now, by the same reasoning as in the first paragraph, there exists some X uv such that X u0v0 appears with unit coefficient in φ(X uv ). By the second paragraph q r0s0,u0v0 = q rs,uv = q r0s ′ 0 ,u0v0 , a contradiction.
2. Let φ be an O p -algebra automorphism of A p and assume that φ(X rs ) / ∈ T p , for some 1 ≤ r ≤ t and 1 ≤ s ≤ m r . By part (3d) of Lemma 5.2, there exist 1 ≤ u ≤ t and 1 ≤ v ≤ m u such that q rs,uv = 1. Let a ∈ A such that X a appears with non-zero coefficient in either φ(X rs ) or φ(X uv ) and let a be minimal with respect to this property. We set these coefficients to be a rs and a uv respectively. Let X r0s0 (respectively X u0v0 ) appear with with coefficient a r0s0 ∈ O × p (respectively a u0v0 ∈ O × p ) in φ(X rs ) (respectively φ(X uv )), note their existence is guaranteed by part (1). Furthermore let X r0s0 appear with coefficient b r0s0 in φ(X uv ) and similarly X u0v0 with coefficient b u0v0 in φ(X rs ).
X a X r0s0 appears with coefficient a uv a r0s0 + a rs b r0s0 in both φ(X uv X rs ) and φ(X rs X uv ) = q rs,uv φ(X uv X rs ). Now 1 − q rs,uv is invertible in k and hence also in O p and so a uv a r0s0 + a rs b r0s0 = 0. Similarly, by comparing coefficients of X a X u0v0 , we have that a rs a u0v0 + a uv b u0v0 = 0. Taking these two equalities together gives v p (a rs ) = v p (a uv ), where v p is the valuation of O p with respect to its unique maximal ideal. This implies b r0s0 and b u0v0 are both invertible. Part (1) of this lemma now gives 1 = q r0s0,r0s0 = q rs,uv , a contradiction.
3. Let φ be an O I -algebra automorphism of A I and assume that φ(X rs ) / ∈ T I , for some 1 ≤ r ≤ t and 1 ≤ s ≤ m r . We define u, v, a, a rs and a uv exactly as in part (2). Without loss of generality, let a rs be non-zero. Note that by part (2), we must have a rs , a uv ∈ 2O I . As in part (2), there must exist some X u0v0 (respectively X r0s0 ) with unit coefficient in φ(X uv ) (respectively φ(X rs )). Let X u0v0 appear with coefficient a u0v0 in φ(X uv ) and b u0v0 in φ(X rs ). We note that by part (1), b u0v0 is not invertible.
We now study the coefficient of X a X u0v0 in φ(X rs X uv ) and φ(X uv X rs ). By part (5) of Lemma 5.2 the only non-zero contributions must come from taking X a in φ(X rs ) and X u0v0 in φ(X uv ) or taking X a1 X u0(v0−1) with unit coefficient in φ(X rs ) and X a2 X u0(v0−1) with unit coefficient in φ(X uv ), where a 1 + a 2 = a. (Note that as b u0v0 is not invertible and a uv ∈ 2O I , b u0v0 a uv = 0 and so we need not consider taking X u0v0 in φ(X rs ) and X a in φ(X uv ).) In particular, the coefficients of X a X u0v0 in φ(X rs X uv ) and φ(X uv X rs ) = q −1 rs,uv φ(X rs X uv ) are the same and, since q rs,uv = 1 in O p , zero. This implies the case of taking X a1 X u0(v0−1) with unit coefficient in φ(X rs ) and X a2 X u0(v0−1) with unit coefficient in φ(X uv ) must make a non-zero contribution in both φ(X uv X rs ) and φ(X rs X uv ).
Let b ∈ A such that X b X u0(v0−1) appears with unit coefficient in φ(X rs ) or φ(X uv ) and let b be minimal with respect to this property. Note that b < a since otherwise X u0(v0−1) itself appears with unit coefficient in either φ(X rs ) or φ(X uv ), contradicting the minimality of b, unless a = b = ∅. In this case X u0(v0−1) appears with unit coefficient in φ(X rs ) but this is a contradiction as, by part (1), q rs,uv = q u0(v0−1),u0v0 and, by part (3a) of Lemma 5.2, q u0(v0−1),u0v0 = 1.
Say X b X u0(v0−1) appears with unit coefficient in φ(X rs ). Then we consider the coefficient of X b X u0(v0−1) X u0v0 in both φ(X rs X uv ) and φ(X uv X rs ) = q −1 rs,uv φ(X rs X uv ). In particular, we consider their images in k. The only non-zero contribution is from taking X b X u0(v0−1) in φ(X rs ) and X u0v0 in φ(X uv ). (Note that by the final part of (1), X u0(v0−1) cannot appear with unit coefficient in φ(X uv )). So the coefficients are equal and non-zero. This is a contradiction as q rs,uv = 1.

Weiss' condition and the main theorem
In this section we prove our main result. Along with the results already proved in this article, our main tool will be an application of Weiss' condition. Weiss' condition is a statement about permutation modules originally stated in [15,Theorem 2] but proved in its most general form in [14,Theorem 1.2]. Proposition 6.1 is a consequence of the condition that was proved in [6,Propositions 4.3,4.4]. We first set up some notation. In the above paragraph we need to be a little careful when we apply the conclusion from the first paragraph. First note that a block C appearing in the direct sum B [D,Z(E)] may not be of the form of a block as described in §3. In particular, the relevant character ϕ C of Z(E) may not be faithful. However, C is naturally Morita equivalent to a block of G/([D, Z(E)] ker(ϕ C )) that will be of the desired form. Secondly we are implicitly using the fact that A slightly more delicate argument is required for the p = 2 case due to the weaker result obtained for p = 2 in Lemma 3.6. When B has a unique simple module we first use part (2a) of Lemma 3.6, part (4) of Lemma 5.2 and part (2) of Lemma 5.4 to apply part (2) of Proposition 6.1 with respect to B, M and Φ(D 1 ). In other words, we may assume that D 1 is elementary abelian. We can now use part (2b) of Lemma 3.6, part (4) of Lemma 5.2 and part (3) of Lemma 5.4 to apply part (2) of Proposition 6.1 with respect to B, M and D 1 . The general p = 2 case now follows exactly as for p > 2.
This time we are implicitly using the fact that [D/Φ(D 1 ), E] = D 1 /Φ(D 1 ). This is required when we reduce to the situation of D 1 being elementary abelian.
In what follows, for any finite group H and α ∈ Aut(H), we denote by ∆α the subgroup {(g, α(g))|g ∈ H} of H × H. If α = Id H , we simply write ∆H. O H (respectively K H ) will signify the trivial OH-module (respectively trivial KH-module). We are now ready to state and prove our main theorem. Proof. We first note that b is source algebra equivalent to a block of the form of B as introduced in §3, where the defect group of b is isomorphic to D and its inertial quotient is isomorphic to L. The fact that we have a Morita equivalence follows from [12,Theorem A] and that this Morita equivalence is in fact a source algebra equivalence from [13, Theorem 6.14.1]. Note that in both of these articles actually an equivalence with a twisted group algebra O α (D ⋊ L) is constructed. However, O α (D ⋊ L) and B are isomorphic as interior D-algebras, for an appropriately chosen B. (See the comments following [10,Theorem 4.2] for a discussion of [12,Theorem A].) Note that E must be a p ′ -group as otherwise G has a normal p-subgroup strictly containing its defect group.