A block theoretic analogue of a theorem of Glauberman and Thompson

If $p$ is an odd prime, $G$ a finite group and $P$ a Sylow-$p$-subgroup of $G$, a theorem of Glauberman and Thompson states that $G$ is $p$-nilpotent if and only if $N_{G}(Z(J(P)))$ is $p$-nilpotent, where $J(P)$ is the Thompson subgroup of $P$ generated by all abelian subgroups of $P$ of maximal order. Following a suggestion of G. R. Robinson, we prove a block-theoretic analogue of this theorem.