1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

3.1. COMMON NORMAL SUBGROUPS, AND THE qrc-LEMMA FOR QTKE-GROUPS 575

Let Y denote a Hall 2' -subgroup of B. As D :::) B by the previous paragraph,

Yv := Y n Dis also Hall in D, so D = YvR = RYv. Further Y:::: B:::: Na(D), so

YR= YYvR= YD =DY =RYvY =RY.

Then R is Sylow in the group YR, and Y normalizes 02 (YR) :::: R.

We claim that T n K :::: R; this is the crucial step ,in showing that R is Sylow

in RK, and hence in completing the proof. Since T is transitive on the groups

Ki, it suffices to show that Ti := T n Ki :::: R ·for some i. Let Qi := 02 (Ki),

To := Nr(Ki), Yi := Y n Ki, and KiTo := KiTo/02(KiTo). Then A :::: T 0 by

B.1.5.4, and as A 1. QH, while Ki is quasisimple or of order 3, we may choose i so

that Ki= [Ki,A]. Next

But if Ki ~ A3 then Pi= Qi E Syl2(Ki) since Ki= 02 (Ki), so that Ti= Pi:::: R

by (*), as claimed.

Suppose next that fji is the natural module for Ki ~ L 2 (2n) with n > 1.

Then by B.4.2.l, the FF*-offender A is Sylow in 'Ki, so that Ti :::: J(R)Qi with

J(R) :::: 02(YiTo). Thus J(R) :::: 02(YT0), so

J(R):::: 02(YT0) n YR:::: 02(YR):::: R,

so Y acts on J(02(YR)) = J(R) using B.2.3.3, and hence again using(*),

Ti = [J(R), Yi]Pi :::: RPi :::: R.

Finally if Ui is .the natural module for Ki ~ A5, then by B.3.2.4, the FF*-

offender A is generated by one or two transpositions. Thus [A, Ti] :::: Rn Ki =: Ri,

so as [A, Ti] 1. Qi, (*) says

Ti= (Rr)Pi = RiPi:::: R.

We have established the claim that T n K :::: R. Since T·is Sylow in H and

K :::)H, TnK is Sylow in K, so R is Sylow in RK, completing the proof of Theorem

3.1.6.. D

The next result is another corollary of Theorem 3.1.1, in the same spirit as

Theorem 3.1.6. Recall that Z is 01(Z(T)), and the Baumann subgroup ofT from

Definition B.2.2 is Baum(T) = Cr(01(Z(J(T)))).

LEMMA 3.1.7. Assume Hypothesis 3.1.5, with J(T) :::: R. Then either

(1) Z:::: Z(H) and Z(M 0 ) = 1, or

(2) There is 1 f= Ro:::: R with Ro :::) (Mo,H).

PROOF. By hypothesis J(T) :::: R = Cr(V). Then J(T) = J(R) and S :=

Baum(T) = Baum(R) by B.2.3.5 with Vin the role of "U". Therefore if J(T) :::) H,

then (2) holds with J(T) in the role of "Ro". Thus we may assume J(T) is not

normal in H,'so His not 2-closed. Hence HE Ua(T) and His des.cribed in B.6.8
by 3.1.3.
Suppose Z :::: Z(H). If Z(M 0 ) = 1 then condusion (1) holds, so we may
assume Z(Mo) f= 1. By Hypothesis 3.1.5, Mo E 7-i(T), and hence Mo E 7-ie by
1.1.4.6. Therefore Z(M 0 ) is a 2-group, so fh(Z(Mo)) :'::: Z :::: Z(H), and hence
conclusion (2) holds with 01(Z(Mo)) in the role of "Ro"·

Thus we may assume that Z 1. Z(H). Let UH:= (zH) and K := 02 (H). As

H =KT, K 1. CH(Z), so Ki CH(UH)· We saw in the previous paragraph that