1008 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L:r(G, T) IS EMPTY26 is Sylow in both Land Ki. Then (Ki, Cr(L)] ~[Ki, Cr(U)] ~ CK 1 (U) =Vi, so
K ~ Ca(Cr(L)) by Coprime Action; therefore Cy(L) = 1 as Ki M = !M(LT).
Let A := 02 (M). As M = LT and L is an L3(2)-block with V = 02(L) byparts (4) and (7) of 14.4.4, A is elementary abelian by C.1.13.1, while m(A/V) ~
dimHi(L/V, V) = 1 by C.1.13.b and I.1.6.4. Thus either (1) holds, or A~ E15 and
T /A is regular on A-V from the structure in B.4.8.3 of the unique indecomposable
A with [A, L] = V. But in the latter case, A = J(T) using 14.4.4.6, and all
involutions in T - Lare in A. However as J(T) =A, Na(A) = M controls fusion
in A by Burnside's Fusion Lemma A.1.35, so a^0 n L = 0 for a EA-L, and then
Thompson Transfer contradicts the simplicity of G.^2
Therefore (1) is established. Now (3) follows from (1) and 14.4.4.5. Then
H =KT= Ki, and (2) holds. D
We are now in a position to complete the proof of Theorem 14.4.3. We will
show G is of G 2 (3)-type in the sense of section I.4, and then conclude G ~ G2(3)
by the classification theorem stated in Volume I as I.4.5.
First by 14.4.4.4 and 14.4.5.1, F*(M) = V ~ E 8 and M/V ~ L3(2). Second
U = 02 (H) by 14.4.5.2, and as d = 4, U ~ Qg by 14.4.2. By 14.4.4.1, K* ~ Eg,
so K = KiK2, where Ki ~ 8L2(3), [Ki, K2] = 1, and Kin K2 = Vi. By 14.4.5.2,
IH: Kl= 2. Further by 14.4.4.2, WY* inverts K*; so WY, and hence also H, actson Ki. Thus G is of G 2 (3)-type, completing the proof of Theorem 14.4.3.
14.4.2. Eliminating the case d > 4. Having established Theorem 14.4.3, we
may assume for the remainder of this section that d > 4; as no quasithin examples
arise, we are working toward a contradiction. In fact d = 8 or 12 since one of cases
(3)-(6) of 14.4.2 holds.
LEMMA 14.4.6. If a* is an involution in H* then either
(1) m([U, a*]) > 2, or
(2) H* ~ 83 x 85 or F*(H) ~ nt(4), and in either case f2 i [U, a*] and
m([U, a*]) = 2.
PROOF. Assume (1) fails. Then by inspection of cases (3)-(6) in 14.4.2, either:
(a) conclusion (5) of 14.4.2 holds, with H* =Hi xH2, where Hi~ 83, H2 ~ 85,
U is the tensor product of the natural module for Hi and the As-module for H2,
and a* is a transposition in H2, or(b) conclusion (3) of 14.4.2 holds, with a* inducing an F 4 -transvection on U.
In case (a), U = Ui EB U2 is the sum of two irreducible H2,-modules with
Co; (T* n H2) = (ui) and ui singular in the orthogonal space Ui. Therefore as
the generator s of f2 centralizes T, s = ih + u 2. However [Ui, a] = (vi) with vi
nonsingular, so s ~ [U, a*], and hence (2) holds.
Similarly in case (b ), f2 is contained in a singular F 4 -point of U, while [U, a*]
is a nonsingular F 4 -point, so again (2) holds. D
LEMMA 14.4.7. U = 02(H) = Q.
PROOF. As U is extraspecial, 02 (HY) = UYD, where D := CH9(UY). Now as
g E NL(Vz), Vz ~UY, so [D, W] ~ Cw(UY). But Cw(UY) ~UY by 12.8.9.5, so
that Cw(UY) = V{'. Therefore if D* -::f 1, either D induces transvections on U with
(^2) Notice here we are eliminating the shadow of Aut(G2(3)).