Glauberman-Thompson normal $p$-complement theorem for $p=2$

Question

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fails for $p=2$ even if the group is $S_4$-free, the argument in the proof of Theorem D of (Glauberman, 1968a) is no longer valid.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.

Answer 1

I think I can answer your third question (if I read it correctly), even if the answer to the first question is positive. This is all stuff from Glauberman, but appears in my book on fusion systems, Section 7.2. Write $Qd(p)=(p\times p)\rtimes SL_2(p)$, so $Qd(2)=S_4$.

A positive characteristic $p$-functor $W$ is a map from finite $p$-groups $P$ to a characteristic subgroup $W(P)$ of $P$, such that $W(P)\neq 1$ if $P\neq 1$, and if $P\to Q$ is an isomorphism then $W(P)$ is mapped onto $W(Q)$. This is called a Glauberman functor if, whenever $P\in\mathrm{Syl}_p(G)$ for a $p$-constrained group that is $Qd(p)$-free, $W(P)$ is normal in $G$ (i.e., a ZJ-type theorem holds).

Theorem 7.21 of my book, which is Theorem 6.6 of Glauberman's 1971 paper, states that if $G$ is $Qd(p)$-free and $W$ is a Glauberman functor, then $N_G(W(P))$ controls fusion in $G$. So if $N_G(W(P))$ has a normal $p$-complement, so does $G$. (There is no restriction on $p$ for this.)

That seems to answer your third question, I think.

(George Glauberman. Global and local properties of finite groups. In Finite Simple Groups (Proceedings of the Instructional Conference, Oxford, 1969), pp. 1–64. Academic Press, London, 1971.)

Answer 2

As regards the last question (if you are asking what I think you are asking), the proof that (for $p$ odd), $G$ has a normal $p$-complement if and only if $N_{G}(ZJ(P))$ has a normal $p$-complement ( where $P$ is a Sylow $p$-subgroup) is reduces to the case that $G/O_{p}(G)$ has a normal $p$-complement. Then it follows that $G$ is $p$-solvable, and that $P$ is a maximal subgroup of $G$, in which case the normal $p$-complement is an elementary Abelian $q$-group. In particular, $G$ has an Abelian Sylow $2$-subgroup. For either definition of $p$-stable ($p$-odd), a group which is not $p$-stable contains a quaternion subgroup of order $8$, so has non-Abelian Sylow $2$-subgroups.