Defect group and numerical defect of a block

For the block bimodule $kGb$, a defect group is a $p$-subgroup $D\le G$ for which $\mathrm{\Delta }D$ is a vertex. Such a subgroup exists by Block bimodule has a diagonal vertex, since conjugating a vertex produces another vertex. Its order is $p^d$ for a unique integer $d\ge 0$, called the numerical defect. The following conjugacy theorem shows that $d$ is independent of the defect group. Defect zero means $D=1$. This definition uses the residue-field bimodule; it asserts no identification of lattice vertices.

Webb, A Course in Finite Group Representation Theory, §§5.2, 11.3, 11.6, 12.3–12.5; especially Lemma 12.4.4 and Theorem 12.4.5, pp.240–241. Local argument and conventions as displayed above.