I asked this question on Math StackExchange yesterday. As suggested by Professor Derek Holt, this question may be more suitable for this site. So I ask this question here again, but more details and discussions are added.
The following is Thompson’s normal $p$-complement theorem:
Let $G$ be a finite group and $S$ a Sylow $p$-subgroup. Suppose that $p$ is odd or $G$ is $S_4$-free. Then $G$ has normal $p$-complement if and only if $C_G(Z(S))$ and $N_G(J_o(S))$ has normal $p$-complement.
The theorem in the original paper (Thompson, 1964) only gives the part of $p$ odd and uses a different Thompson subgroup $J_r$ in terms of the maximal rank of abelian subgroups, whereas $J_o$ is defined in terms of the maximal order of abelian subgroups. Recall that a group is called $H$-free if no section of $G$ is isomorphic to $H$. In many later references, “$S_4$-free” is added in the statement. See for example for both points Corollary 5 of (Glauberman, 1968b) and p. 45 of (Glauberman, 1971).
Another famous normal $p$-complement theorem is the one attributed to both Glauberman and Thompson:
Let $G$ be a finite group and $S$ a Sylow $p$-subgroup. Suppose that $p$ is odd. Then $G$ has normal $p$-complement if and only if $N_G(Z(J_o(S)))$ has normal $p$-complement.
First question: At first glance, I wonder if the $p=2$ part is valid for Glauberman-Thompson $p$-nilpotency criterion if the hypothesis “$S_4$-free” is added.
But I could not find any $S_4$-free examples. I looked then more carefully into the proof of the theorem, namely Theorem D of (Glauberman, 1968a). As indicated by the proof or p. 41 of (Glauberman, 1971), Glauberman-Thompson $p$-nilpotency criterion follows from Glauberman’s ZJ-theorem, namely Theorem A of (Glauberman, 1968a). This leads to the following two question.
Second question: Since Glauberman’s ZJ-theorem fails for $p=2$ even if the group is $S_4$-free, the argument in the proof of Theorem D of (Glauberman, 1968a) is no longer valid. It is logically reasonable to guess that counterexamples to Glauberman’s ZJ-theorem may provide some counterexamples for the statement in the first question. Recall that there is indeed a counterexample given for Glauberman’s ZJ-theorem in $S_4$-free groups when $p=2$. See Example 10.2 of (Glauberman, 1968a). Does this example provide a counterexample?
Along the other direction of the second question, we may ask the following.
Third question: If the answer to the first question is negative, then can we do that with another functor like the one given by Stellmacher, that is, does the $p=2$ case follows immediately from any functor that admits a ZJ-type theorem for $S_4$-free groups of characteristic $2$?
My initial question are the above three. The following question is just a minor one.
Forth question: In p. 41 of (Glauberman, 1971), Glauberman mentioned: “The following results are proved in Section 8.1 and 8.2 of (Gorenstein, 1968).” Here “the following results” refers to Glauberman’s ZJ-theorem and Glauberman-Thompson $p$-nilpotency criterion. As for the ZJ-theorem, it is understandable since the definition of $p$-stability are different in (Glauberman, 1968a) and (Glauberman, 1971), where the latter one is the most frequently used nowadays. But it seems that the statement of the $p$-nilpotency criterion are the same in (Glauberman, 1968a) and (Glauberman, 1971). Why he says it is proved by Gorenstein? Is it just a style of writing? To be more precise, is that because, since the criterion follows from the ZJ-theorem, the proof of the criterion naturally changes accordingly, even if the statements are the same?
References:
Glauberman, G. (1968). A characteristic subgroup of a $p$-stable group. Canad. J. Math., 20:1101-1135.
Glauberman, G. (1968). Weakly closed elements of Sylow subgroups. Math. Z., 107:1-20.
Glauberman, G. (1971). Global and local properties of finite groups. M.B. Powell and G. Higman, eds. Finite Simple Groups (Proc. Instructional Conf. Oxford, 1969). London: Academic Press, 1-64.
Gorenstein, D. (1968). Finite groups.
Thompson, J. G. (1964). Normal $p$-complements for finite groups. J. Algebra, 1:43-46.