= Trace criterion for defect groups
For a $p$-subgroup $D$ of a finite <group> $G$, set $I_D=\operatorname{Tr}_D^G((kG)^D)$, a <transfer ideal of conjugation-fixed elements> in $Z(kG)$. A <block of a group algebra> idempotent $b$ lies in $I_D$ if and only if a <defect group of a block> of $b$ is conjugate to a subgroup of $D$. In particular, defect groups can equivalently be defined as minimal subgroups $D$ with $b\in I_D$.
Here is a proof using the <Brauer morphism>. A subgroup of smallest order with $b\in I_D$ exists, since a <Sylow subgroup> $P$ gives $b=\operatorname{Tr}_P^G(b/[G:P])$. Write $b=\operatorname{Tr}_D^G(a)$ and replace $a$ by $ba$. If $\operatorname{Br}_D(b)=0$, then $ba$ belongs to the kernel of the Brauer morphism, which is the sum of proper-subgroup relative traces. Hence $b\in\sum_{Q<D}I_Q$. The algebra $bZ(kG)$ is a commutative <Artinian ring> and a <local ring>, so multiplying this equality by $b$ forces some ideal $bI_Q$ to contain its identity $b$. This contradicts minimality. On the other hand, $\operatorname{Br}_E(I_D)=0$ unless $E$ is conjugate into $D$: the $E$-action on $G/D$ has no fixed coset otherwise, and nonfixed orbit sizes vanish in characteristic $p$. Thus these minimal subgroups are precisely the maximal subgroups with nonzero Brauer image. They are all conjugate. Trace transitivity proves the stated criterion for larger $D$.
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