= Defect groups are centralizer-conjugate Sylow intersections
{title2=$D=P\cap{}^cP,\quad c\in C_G(D)$}
Let $P$ be any <Sylow subgroup> containing a <defect group of a block> $D$. Regard the <block of a group algebra> as a <permutation module> summand for $G\times G$, acting by left and right multiplication. Its vertex is $\Delta D$. Restricting to $P\times P$ retains a summand with this vertex. Transitive <permutation modules> for a <finite p-group> are indecomposable, with their point stabilizers as vertices. Thus some orbit in $G$ has stabilizer $\Delta D$ at a suitably chosen point $c$. This means $c$ centralizes $D$ and $P\cap{}^cP=D$. Choosing $P$ to contain a Sylow subgroup of $N_G(D)$ shows that $D=O_p(N_G(D))$; every normal <p-subgroup> of $G$ is contained in both Sylow factors and hence in $D$.
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