2.4. THE CASE WHERE rg IS NONEMPTY 547PROOF. By 2.4.27.1, IT : SI = 2, so (2) holds by 2.3.7.l. By 2.4.27, R =
J(S) =To x UoU 0 , where To:= Cs(Lo), and Q =To x U 0 , so R = QQx. Thus (1)
holds, and (3) follows from 2.4.27.2. By 2.4.27.4, R =To x U 0 U 0 and Lo = [L 0 , U 0 ],
so (4) holds. Further as Lf = L2, H/Q ~ S 3 wr Z 2 , and as R = J(S) acts on L 1and Q:::; R, R is Sylow in RLo = NH(L1) of index 2 in S, so (5) holds. D
REMARK 2.4.31. In the remainder of the proof of Theorem 2.4.1, we are again
faced with a shadow of an extension of L4(3), but now approached from the point
of view of a 2-local with two A 3 -blocks. We will construct the centralizer of the
involution z 2 defined below, as a tool for eventually obtaining a contradiction to the
absence of an A5-block in any member of r 0. In PDci(3), z 2 is an involution whose
commutator space on the orthogonal module is of dimension 2 and Witt index 0,
and whose centralizer has a component n4(3) ~ L 2 (9) ~ A 6.
Now let (zi) = Cu; (R). Then by 2.4.30, (z1, z2) = 1>(R) :::;! T and L3-i :::;
Ca(zi) =: Gi, so Gi i. M. Since Q :::; R, we conclude by 2.3.8.5c that Co 2 (M) (R) :::;
R. Then since IS : RI = 2 and R = J(S) by 2.4.30.1, the first sentence of 2.3.8.5b
says RE (3. So since Li i. M, we conclude as usual from the definitions in Notation
2.3.4 and Notation 2.3.5 that (R, L3-iR) E U( Gi) and Gi E r. Next zf = z 2 , soz := z 1 z 2 generates Z(T)n1>(R), and replacing x by xs if necessary, we may assume
x E Gi, for i = 1 and 2. Let S1 := R(x). Then IT: Sil= 2 =IT: SI, so by 2.3.7.1,
Gi Er* and S1 E Syl2(Gi)·.
Observe that F*(G 2 ) # 02(G2): For otherwise by 2.3.8.4 and 2.4.29, G2 =CM(z 2 )Ko, where Ko is the product of two A3-blocks. But R = J(S1), so applying
2.4.30.4 to KoS1, CiJ!(R)(Ko) = 1, contradicting Z2 E ciJ!(R)(Ko).
Next 02(L) ~ E4 centralizes O(G2) by A.l.26, so z E (z2)02(L):::; Ca 2 (0(G2)),
and hence O(G2) = 1 since z inverts O(G2) by 2.3.9.5. Thus as F*(G2) # 02(G2),
there exists a component K of G 2 , and K is described in 2.3.9.7. By 2.3.9.6,K = [K, z], so L is faithful on K since z E (z2)L.
Recall S1 E Syb(G2) and IS1: RI= 2 with R = Ns 1 (L); therefore Auts 1 (K) E
Syl2(Auta 2 (K)) with IAuts 1 (K): NAuts 1 (K)(AutL(K))I:::; 2. Further we saw L
is faithful on K, so Auh(K) ~ A 4. Inspecting the 2-locals of the automorphism
groups of the groups K listed in 2.3.9. 7 for such a subgroup, and recalling 0( G2) =
1, we conclude that K is one of A5, A5, A1, As, L2(7), L2(17), L3(3), or M11.
Moreover if LK is the projection of L on K, then as IS1 n K : Ns 1 nK(L)I :::; 2
(since Lis irreducible on 02(L) ofrank 2), 02(LK):::; NK(L), and then 02(LK) =
[0 2 (LK), L] = 02(L) = U. As S1 centralizes z and z2, S1 centralizes z1 = zz2 E
U:::; K, so S 1 acts on Kand hence K :::;! 02 by 1.2.1.3. If K ~ A5 or A1, then
U :::;! S1, contradicting x ~ Nr(U). If K is As, then Lis an A4-subgroup moving
4 of the 8 points permuted by K, so z1 is not 2-central in K, a contradiction. If
K is L 3 (3), M 11 , or L2(17), there is XK E S1 n K with QXK # Q, so we may take
x = xK; but now IQx: CQx(Q)I = 2, contradicting 2.4.30.1 which shows this index
is 4. Therefore:
LEMMA 2.4.32. G2 Er* and L:::; K ~ A6 or L2(7).
Next zf^2 = zf since A5 and L 3 (2) have one class of involutions; so by a Frattini
Argument, G2 = KCa 2 (z1) = KCa 2 (z) = KM2, where M2 :=Mn G2. As G2 E
r, F(M2) = 02(M2) by 2.3.9.4. Then as Ca 2 (K):::; M2, F*(G2) = K02(G2). In
particular: