i3.8. FINISHING THE TREATMENT OF A 6 95ia contradiction. Thus P i M, so by minimality of H, H = PLiT and LiT is
irreducible on P /i!!(P). As Pi M, X -=/-P.
As u; E Q(H, UH), p = 3 or 5 by D.2.13.1.
Suppose first that p = 3. If P is of order 3, then as m 3 (H) ~ 2 and Pi M,
03(H) = P x Li~ Eg; hence H ~ 83 x 83 as VO: is nontrivial on P. Therefore
VO:= 02(LiT*) ~ Z2, so m(V;) = m(U;) = 1. Also Li ::::] H, so UH= [UH, Li],
and hence m(UH) = 2m 2::: 4, where m := m([UH,u;]). Now by 13.8.10.1, we
have symmetry between ry and ryi, so UH does not induce transvections on Ury/Ai.
Hence Vi ~ Ury by 13.8.10.2. Further H = LiT(U!/), so by symmetry, Gry =
Grym_ 1 (Ui;-r), and hence m(V;) > 1by13.8.11.1, contradicting IV;I = 2.
Therefore P ~ Eg or 31+^2. Suppose Li i P. As LiT is .. irreducible on
P /if!(P), H induces SL 2 (3) or GL 2 (3) on P /i!!(P). So in particular if P ~
Eg, then m([UH, P]) 2::: 8; as u; E Q(H, UH), this contradicts D·.2.17. Hence
P ~ 3i+^2 , so that m 3 (LiP) > 2, contradicting H an SQTK-group. Therefore
Li ~ P, so Li < P as X -=/-P. Then as LiT is irreducible on P /i!!(P),
P ~ 31+^2 and .Li = Z(P), so that L/02(L) ~ A5. Then 02(LiT) =Cr· (Li) is
of2-rank at most 1, so m(V;) = 1. As u; E Q(H,UH), u; inverts P/if!(P) by
D.2.17.4. Now we obtain a contradiction as in the previous paragraph.
We have reduced to the case p = 5. As u; E Q(H, UH) and His minimal, we
conclude from D.2.17 that P =Pix··· x Ps withs~ 2, and [P, UH]= Ui EB·· ·EB Us
with Pt ~ Z 5 , where Ui := [Pi, UH] is of rank 4. Ifs= 1 then u; ~ Z 2 , while if
s = 2, then either u; ~ Z2 with [u;, P2] = 1, or u; = Bi x B2 with Bi ~ Z2
centralizing P3-i· However if u; is of order 4 then m(UH/CuH (Ury)) = 4 = 2m(u;),
and so F.9.16.2 shows that m(UH/ DH)= 2 and u; acts faithfully on DH as a group
of transvections with center Ai. Then m([UH, u;]) ~ 3, whereas this commutator
space has rank 4 since s = 2.
Therefore m(U;) = 1, so as before we have symmetry bewtween 'Yi and ry by
13.8.10.1; and as LiT is irreducible on P*, Gry = G'Ym- 1 (ui;-r). As p = 5, no
element of H* induces a transvection on UH by G.6.4; hence we conclude from
13.8.11.2 that m(Vry•) > 1. In particular as v; is faithful on P, P is not cyclic,
so s = 2 and v; = u; x B2 with B2 ~ Z2 centralizing PJ'.
Let CH := CuH(Ury) and PH := CuH(Ury)· By 13.8.10.1, m(UH/DH) = 1,
and by F.9.13.6, [Ury,DH] ~ A1. Thus if FH i DH, then UH = DHFH, so
[UH, u;J ~Ai, contrary to m([UH, u;]) = 2. Hence FH ~DH. Then by F.9.13.6,
[FH, Ury] ~Vin [DH, Ury] ~Vin Ai= 1
and hence U2 ~ FH = CH. Then [U2, Vry] ~ [CH, Vry] ~ Ai by 13.7.3.7, with
Ai = [DH, Ury] ~ Ui. On the other hand, 1 -=I-[U2, B2] ~ [U2, Vry] n U2 ~Ai n U2,
contrary to Ui n U2 = 0. D
By 13.8.13 and 13.8.12.2:LEMMA 13.8.14. Let X := 02 (02,F(H)); then one of the following holds:
(a) X = 1.
{b) L/02(L) ~ A6 and X =Li.
(c) L/02(L) ~ A5 and X =Li,+·