12.4. SOME FURTHER REDUCTIONS 815LEMMA 12.4.8. r( G, V) > 1.
PROOF. Assume that r(G, V) = 1. Then there is a hyperplane U of V with
Ca(U) <J;. Na(V). Let Gu := Ca(U), Mu := CM(U), and Lu := NL(U). Then
Zv <J;. U as Ca(Zv) = Ca(z) = M.
Now consider some hyperplane Uo of U, and set Gu 0 := Ca(Uo) and Mu 0 :=
CM(Uo); then Gu :::; Gu 0 , so also Gu 0 <J;. M. As Zv <J;. U and. m(V/U) = 1,
also Zv <J;. Uo and m(V/UoZv) = 1. For any involution t E M, Zv :::; Cv(f) and
m(V/Cv(f)) ~ 2 (cf. B.4.8.2 and B.4.6), sot does not centralize U or U 0. Thus U
and Uo lie in the set r of Definition E.6.4, and we may apply appropriate resultsfrom that section. In particular by E.6.5.1, Q is Sylow in Gu 0 and Gu. Also Mu 0
centralizes the quotients of the series V > UoZv > U 0 > 1, so by Coprime Action',
Mu 0 is a 2-group. But we just observed that Mu 0 does not contain involutions, so
we conclude that Mu 0 = CM(V), and hence also Mu= CM(V).
Now if L ~ L3(2), then Tis regular on hyperplanes not containing Zv, so Uis determined up to conjugation under T, and Lu~ Frob21- On the other hand, if
L ~ G2(2)', then by Theorem 2 in [Asc87], L has two orbits on hyperplanes not
containing Zv, exhibited by conclusions (3) and (4) of Theorem 3 in [Asc87], and
given by representatives U1 and U2, where Lu 1 ~ PSL2(7) and Lu 2 ~ Qs/31+^2.
When LT= G2(2), the stabilizers in LT are twice as large. In each case NLT(U) is
maximal in LT, but not of index 2; further Lu contains Xu of order 3 faithful on U.
Thus if F*(Gu) = 02 (Gu), then the hypotheses of lemma E.6.14 are satisfied with
U and LT in the roles of "W" and "Mo", so by that lemma, Gu= Ca(U) :::; M,
contrary to our assumption.This contradiction shows that Gu tJ. 1-{e. Suppose next that there is a com-
ponent K of Gu. Then K is described in E.6.8, and in particular K <J;. M. Now
Kn M :::; Mu = CM(V), so that [V, Kn M] = 1. If case (1) of E.6.8 occurs,
this forces n = 1, so that K ~ L 3 (2); we regard this group as L 2 (7), and treat it
with the groups L 2 (p) arising below. In particular K is not a Suzuki group. The
existence of Xu of order 3 faithful on U and an appeal to A.3.18 eliminates all cases
of 3-rank 2-namely (2)-(4) of E.6.8, and all cases of E.6.8.5 except K ~ L 2 (p), pa
Fermat or Mersenne prime. Notice now that the case U = U2 for L ~ G2(2)' cannotarise: For in that case there is Yu ~ 31+^2 faithful on U, so as m3(Na(U)) :::; 2,
Gu is a 3^1 -group, whereas K is not a Suzuki group. In the remaining two cases
choose l E NL (Xu) - Lu with Z^2 E Lu n Lb, and choose the hyperlane Uo of U
to be U 0 :=Un U^1 • As l acts on Xu and Xu is faithful on U, Xu acts faithfullyon Uo. We saw Q E Syl2(Gu 0 ), so K :::; Ko E C(Gu 0 ) by 1.2.4. Since K ~ L2(q)
rather than SL 2 (q), K centralizes O(Gu 0 ) by A.1.29. Now by I.3.1, K is contained
in the product of a U-orbit of 2-components of Gu 0 , so as K centralizes O(Gu 0 ),
we conclude those 2-components are ordinary components. Hence K:::; E(Gu 0 ), so
Ko is a component of Gu 0 • As Xu is faithful on Uo, we may argue as before that
Ko~ L 2 (q) for q a Fermat or Mersenne prime. But no proper embedding K <Ko
of these groups appears in A.3.12, so we conclude Ko = K is also a component ofCa(Uo). Indeed K = 0
31
(E(Gu 0 )) since m3(Na(Uo)):::; 2 and Xu is faithful on Uo.
But Z^2 E LunLtr so that Uo = U5. Hence K^1 = 0
31(E(Guz)) o = 03 ' (E(Gu 0 )) = K.
Then as K:::; Gu, K centralizes UU^1 = V, contrary to K <J;. M.
Therefore F*(Gu) = F(Gu). As Gu r/:. 1-{e, we conclude Ou := O(Gu) -=f. 1.
Again let U 0 denote a hyperplane of U. By 1.1.6, the hypotheses of 1.1.5 are