Normal p core lies in every block defect group

For finite $K$, the product of two normal $p$-subgroups $N_1,N_2$ is normal and is a subgroup; its order is $\mathrm{\mid }{N}_1\mathrm{\mid }\mathrm{\mid }{N}_2\mathrm{\mid }\mathrm{/}\mathrm{\mid }{N}_1\cap N_2\mathrm{\mid }$, a power of $p$. The order formula follows because each element of the product has exactly $\mathrm{\mid }{N}_1\cap N_2\mathrm{\mid }$ representations as $n_1{n}_2$. There are finitely many subgroups of $K$, so their finite product defines the largest normal $p$-subgroup $O_p(K)$, including the trivial subgroup. If $T$ is Sylow and $N⊴K$ is a $p$-subgroup, the same order formula makes $NT$ a $p$-subgroup containing $T$; maximality of $T$ forces $NT=T$. Thus $N\le T$.

Choose a Sylow $P$ containing $D$ by [F2]. By [F1], $D=P\cap hP{h}^{-1}$ for some $h$. Apply step 1.1 to both Sylow subgroups to obtain $O_p(G)\le P\cap hP{h}^{-1}=D$.