= Regular defect-group bimodule inside a block
{title2=$kD\mid\operatorname{Res}_{D\times D}^{G\times G}(kGb)$}
For a block idempotent $b$ with defect $D$, the <Brauer quotient of a module> of its <modular block> <bimodule> at $\Delta D$ is nonzero: on the fixed basis $C_G(D)$, the block projection acts by $\operatorname{Br}_D(b)$. Restriction to $D\times D$ splits into transitive <permutation modules>; their stabilizers have order at most $|D|$. A nonzero quotient at $\Delta D$ therefore forces a summand with stabilizer exactly $\Delta D$, which is the regular $kD$ <bimodule>. Tensoring on the right with any $kD$-module $U$ gives $U\mid\operatorname{Res}_D^G(b\operatorname{Ind}_D^G U)$. Consequently <finite representation type> of the <modular block> forces finite representation type of $kD$.
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