Thompson's normal $p$-complement theorem states that
For an odd prime $p$ dividing the order of a finite group $G$, if $N_G(J(P))$ and $C_G(Z(P))$ has normal $p$-complements, for $P$ a Sylow $p$-subgroup $G$, then so does $G$.
Now, $J(P)$ stands for the Thompson subgroup which is generated by the abelian subgroups of $P$ of maximal rank. How did Thompson know that $J(P)$ plays a central role to prove Frobenius' conjecture? Are there any earlier results on $J(P)$ which gives Thompson the idea that this theorem holds? Or shortly, why did Thompson define $J(P)$?