Every local full-defect block induces to a global block of defect D

We first record the elementary normalizer condition: if $P<T$ are finite $p$-groups, let $P$ act on $T\mathrm{/}P$ by left multiplication. Nonfixed orbit sizes are divisible by $p$, and $[T:P]$ is divisible by $p$. The fixed points are $N_T(P)\mathrm{/}P$ and include the identity coset, so their positive cardinality is divisible by $p$; hence $N_T(P)>P$. Now if $x$ has nonzero coefficient in $e$, F1 puts $x\in C_G(D)$. By F7 take a Sylow subgroup $T$ of $C_G(x)$ containing $D$. If $T>D$, its subgroup $R=N_T(D)>D$ lies in $N$ and centralizes $x$. The coefficient of $x$ survives ${\mathrm{Br}}_R(e)$ by F6, contradicting F3 for the local block. Thus $D$ is Sylow in $C_G(x)$ for every support element $x$ of $e$.

Suppose $y=gx{g}^{-1}$ also centralizes $D$, with $x$ in that support. Both $gD{g}^{-1}$ and $D$ are Sylow in $C_G(y)$: the first by step 1.1, and the second by equal order and $D\le C_G(y)$. F7 supplies $c\in C_G(y)$ with $cgD{g}^{-1}{c}^{-1}=D$. Then $cg\in N$ and $(cg)x(cg{)}^{-1}=y$. Since $e$ is central in $kN$, its coefficients at $x$ and $y$ agree. Therefore its coefficients are constant on every intersection of a $G$-conjugacy class with $C_G(D)$, with zero throughout intersections missing the support. Give each full $G$-class that common coefficient, zero for a class disjoint from $C_G(D)$. This finite class sum is $a\in Z(kG)$ with ${\mathrm{Br}}_D(a)=e$ by F6.

By F5 the image of $Z(kG)$ is a finite commutative quotient algebra. Its idempotent $e$, present by step 2.1, lifts by F4 to a central idempotent $u\in kG$. Write $u$ as a sum of distinct global primitive block idempotents $f_j$ using F8. Their Brauer images are orthogonal idempotents, central in $kN$ because $N$ normalizes $D$, and sum to $e$. Since $e$ is primitive in $Z(kN)$, exactly one image equals $e$ and all others are zero. Let $f$ be that block idempotent and $B=kGf$. F2 applies because $C_G(D)\le N$ and gives ${\lambda }_{b^G}(f)={\lambda }_b(e)=1$. F8 forces $b^G=B$.

Its Brauer image at $D$ is nonzero. If $S>D$ had ${\mathrm{Br}}_S(f)\mathrm{\ne }0$, put $R=N_S(D)>D$ by step 1.1. Then $D\le R\le N$. Since $C_G(S)\subseteq C_G(R)\subseteq C_G(D)$, F6 gives ${\mathrm{Br}}_R(e)={\mathrm{Br}}_R(f)\mathrm{\ne }0$: a nonzero coefficient retained at $S$ is still retained at $R$. This contradicts F3 for the local block $e$. Thus no such $S$ exists, and F3 proves $D$ is a global defect group of $B$. For $D=1$ the normalizer is $G$, and the proof gives the original defect-zero block; when no local defect-$D$ block exists the universal assertion has no inputs. All lifting and subgroup selections here concern finite sets, so no AC or stronger Green restriction theorem is used.