Our notation and terminology are standard and tend to follow the notation and terminology of [2, 3, 4].

Let k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N⁢⊴⁢G, and let c∈𝐵⁢ℓ(kN)G. Thus we are in the situation of fusion systems of blocks and Clifford theory of [3, Section 8.6]. This section is devoted to a study of fusion systems arising from the G-algebra (k⁢N)⁢c and the results of [2, Section 5.9].

Thus, by [3, Definition 8.6.1], a (G,N,c)-Brauer pair is a pair (R,fR), where R is a p-subgroup of G and fR∈𝐵⁢ℓ⁡(k⁢CN⁢(R)⁢𝐵𝑟R⁡(c)). The results of [3, Section 8.6] imply that the set of (G,N,c)-Brauer pairs is a (finite) poset under a defined inclusion. Moreover, by [3, Definition 8.6.2 and Theorem 8.6.3], if (P,fP) is a maximal (G,N,c)-Brauer pair, then ℱ(P,fP)⁢(G,N,c) is a fusion system on P and (P⁢N)/N∈𝑆𝑦𝑙p⁢(G/N). Note that all maximal (G,N,c)-Brauer pairs are G-conjugate.

Let Q be a defect group of c∈𝐵⁢ℓ⁡(k⁢N). Thus 𝐵𝑟⁡(c)∈𝐵⁢ℓ⁡(k⁢NN⁢(Q)) with defect group Q and, [4, Theorem 5.5.15], 𝐵𝑟Q⁡(c) corresponds to an orbit Γ of NH⁢(Q) on 𝐵⁢ℓ⁡(k⁢CN⁢(Q)) such that Γ is the set of blocks of k⁢CN⁢(Q) that are covered by 𝐵⁢ℓQ⁡(c) and 𝐵𝑟Q⁡(c)=∑γ∈Γγ is an orthogonal sum of blocks of k⁢CN⁢(Q). Thus {(Q,γ)∣γ∈Γ} is the set of (G,N,c)-Brauer pairs with first component Q. Let fQ∈Γ.

Let (R,fR) be a maximal (G,N,c)-Brauer pair. Then, as in [3, Section 8.6], R∩N is a defect group of c∈𝐵⁢ℓ(kN)G. Then there is an x∈N such that (R,fR)x=(Rx,fRx) is maximal (G,N,c)-Brauer pair that contains (Q,fQ).

Let (P,fP) be a maximal (G,N,c)-Brauer pair that contains (Q,fQ). Thus P∩N=Q and Q⁢⊴⁢P⁢⊴⁢NG⁢(P)≤NG⁢(Q).

Set

so that Q⁢⊴⁢H, CN⁢(Q)⁢⊴⁢H and (Q,fQ) is an (H,CN⁢(Q),fQ)-Brauer pair. Thus if (U,fU) is a maximal (H,CN⁢(Q),fQ)-Brauer pair that contains (Q,fQ), then ℱ(U,fU)⁢(H,CN⁢(Q),fQ) is a fusion system such that

Set

Clearly, (Q,fQ)∈Φ(G,N,c∣(Q,fQ)).

Similarly, Φ(H,CN(Q),fQ∣(Q,fQ)) is defined and

As suggested by [3, Lemma 8.6.4]:

Let S,T be p-subgroups of G such that Q≤S≤T, and let fS∈𝐵⁢ℓ⁡(k⁢CN⁢(S)) and fT∈𝐵⁢ℓ⁡(k⁢CN⁢(T)) be such that

Thus

Thus we have:

Let (U,fU) satisfy (a) and (b) of Corollary 4. Then ℱ(U,fU)⁢(G,N,c) and ℱ(U,fU)⁢(H,CN⁢(Q),fQ) are fusion systems by [3, Theorem 8.6.3].

Our main result is: Let 𝒞 denote the full subcategory of ℱ(U,fU)⁢(G,N,c) of objects in Φ(G,N,c∣(Q,fQ)), and let 𝒟 denote the full subcategory of

of objects in Φ(H,CN(Q),fQ∣(Q,fQ)). Thus we have:

Our last results relate to blocks.

Let B∈𝐵⁢ℓ⁡((k⁢G)⁢c). By [1, Theorem], there is a unique

Here β⁢𝐵𝑟Q⁡(c)=β and 𝐵𝑟Q⁡(c)=∑γ∈Γγ. Since CN⁢(Q)⁢⊴⁢NG⁢(Q), we have Γ⊆𝐵⁢ℓ⁡(k⁢CN⁢(Q)), and so β covers fQ. Since H=NG⁢(Q,fQ), there is a unique b∈𝐵⁢ℓ⁡((k⁢H)⁢fQ) such that 𝑇𝑟HNG⁢(Q)⁡(b)=β. Hence (𝑇𝑟HNG⁢(Q)⁡(b))G=B.

Here, by [4, Corollary 5.5.6], we have fQ∈𝐵⁢ℓ(kQCN(Q))H. Thus b is an (H,Q⁢CN⁢(Q),fQ)-Brauer pair since b⁢fQ=b. We assert that Q is a defect group of fQ∈𝐵⁢ℓ⁡(k⁢Q⁢CN⁢(Q)). Indeed, 𝐵𝑟Q⁡(c)∈𝐵⁢ℓ⁡(k⁢NN⁢(Q)) with defect group Q. Here 𝐵𝑟Q⁡(c)=∑γ∈Γγ is an orthogonal decomposition of blocks of k⁢CN⁢(Q) such that fQ∈Γ. Also if γ∈Γ, then γ∈𝐵⁢ℓ⁡(k⁢Q⁢CN⁢(Q)) by [4, Corollary 5.5.6] and Q⁢CN⁢(Q)⁢⊴⁢NN⁢(Q). Thus 𝐵𝑟Q⁡(c) covers γ for any γ∈Γ. Hence we have that Q∩(Q⁢CN⁢(Q))=Q is a defect group of γ for any γ∈Γ. Thus Q is a defect group of fQ∈𝐵⁢ℓ⁡(k⁢Q⁢CN⁢(Q)), where Q⁢CN⁢(Q)⁢⊴⁢H, fQ∈𝐵⁢ℓ(kQCN(Q))H.

By Lemma 7, there is a defect group D of b such that D∩Q⁢CN⁢(Q)=Q and a block fD of CQCN((Q)⁢(D) such that (Q,fQ)≤(D,fD) as (H,Q⁢CN⁢(Q),fQ)-Brauer pairs. Here CQ⁢CN⁢(Q)⁢(D)≤CQ⁢CN⁢(Q)⁢(Q)=CN⁢(Q), and so

Thus 𝐵𝑟D⁡(fQ)⁢fD=fD and (D,fD) is an (H,CN⁢(Q),fQ)-Brauer pair such that (Q,fQ)≤(D,fD) as (H,CN⁢(Q),fQ)-Brauer pairs.

Here D is a defect group of b and B. Let 𝒞⁢(D,fD) denote the full subcategory of 𝒞 of objects (S,fS) of 𝒞 such that (S,fS)≤(D,fD) as (G,N,c)-Brauer pairs, and let 𝒟⁢(D,fD) denote the full subcategory of 𝒟 of objects (T,fT) of 𝒟 such that (T,fT)≤(D,fD) as (H,CN⁢(Q),fQ)-Brauer pairs.

Recall that D is a defect group of b and B. Thus, by choosing a maximal (H,CN⁢(Q),fQ)-Brauer pair (U,fU) such that

as (H,CN⁢(Q),fQ)-Brauer pairs and applying Theorem 6, we conclude: