# On Thompson's normal $p$-complement theorem

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 $J(P)$?

• Very nice question, but probably it can only be answered by himself or people that have been working closely with him. Apr 26 '17 at 8:06
• For clues I would read Gorenstein's paper at the Santa Cruz conference, p17 T10. Apr 26 '17 at 8:10
• @ancientmathematician can you give a link or something? Because I wasn't able to find the paper. Apr 29 '17 at 6:59
• books.google.co.uk/… Apr 29 '17 at 7:24

I think the answer is that the definition resulted from experiment and trial and error. Thompson was looking for a single subgroup of $P$ for which the normalizer would determine whwther or not $G$ has a normal $p$-complement.
In the version of his theorem that you state, he does not quite achieve this, because he requires also that $C_G(Z(P))$ has a normal $p$-complement. But there is an improved version of the result in Theorem 3.1, Chapter 8 of Gorenstein's book "Finite Groups", in which the aim is achieved.
The definition of $J(P)$ there is slightly different from the one you give, and probably came later. It is defined as the subgroup of $P$ generated by all abelian subgroups of maximal order (rather than maximal rank), and the theorem, which is attributed to Glauberman and Thompson, states that, for odd primes $p$, $G$ has a normal $p$-complement if and only if $N_G(Z(J(P)))$ does.
• Thank you for the answer. But how one can notices that the subgroup that he needs to define is $J(P)$ by experiment? Using GAP or some other computer algebra systems, it seems possible. But back then, it seems much more harder. Apr 29 '17 at 7:01