1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

also V9 = [V9,L 2 ]. But then V9 = [R 2 ,L 2 ], contradicting V^9 ::; Ri. This final

contradiction completes the proof of 11.5. 7. D

With 11.5.7 in hand, we are finally in a position to obtain a contradiction to
11.5.3.3.
PROPOSITION 11.5.8. K =Li.
PROOF. Assume that Li < K; then we may take Hi :=KT to be our chosen
member of 1-li. Observe that Hypothesis F.7.6 is satisfied with LT, Hi, LiT in the
roles of "Gi, G 2 , Gi, 2 ". Adopt the notation of section F.7, and in particular let
b := b(r, V). By 11.5.7, U := (VH^1 ) is elementary abelian, sob> 1 by F.7.7.2.
Assume first that bis even. Then by F.7.11.2, there is g E G with 1 # [V, V^9 ] ::;
V n V9, and by F.7.11.5 with the roles of 'Yo and 'Y reversed, we may choose V^9 ::;

02(Gi, 2 ) = Ri. Then inspecting the subgroups of Ri acting quadratically on V,

either

(i) Vi = [V, V9] is a 1-dimensional F q-subspace of V and V^9 , or

(ii) L ~ Sp4(q) and (conjugating in Li ifnecessary) [V, V^9 ] = V2, and [Vs, V^9 ] =

Vi is a 1-dimensional F q-subspace of V and V9.

In either case, as L9 is transitive on 1-dimensional F q-subspaces, we may choose

g E Gi. Then 11.5.7 contradicts our choice of 1 =F [V, V9].

Sob is odd. Pick 'Y at distance bas in F.7.11, choose a geodesic

/'o, 'Yi, ... , /'b := 'Y
in r, and choose g so that 'Yig = 'Y· Thus V i. 02(Hf) and as 'Yi is on the
geodesic, [U, U9] ::; Un U9 by F.7.11.1. By 11.5.6.6, Ui := Un 02(Hf) and

Uo := U9 n 02 (Hi) are of index at most 2 in U and U9, respectively. Further

Vin Vj_^9 = 1-or else by 11.4.2, K = Kg, so [U, 02 (Hf)] ::; [U, K9] = [U, K] ::; U,

contradicting Vi. 02 (Hf). Thus [Ui, U 0 ] ::; Vin V{ = 1, so U 0 centralizes Ui n V

of corank at most 1 in V. However by 11.2.2.3, s(G, V) = m(M, V) = n > 1, so

Uo centralizes V by E.3.6. Then as 1 =F [V, V9] ::; [V, U9], V induces a group of

transvections on U^9 with axis Uo. As Vi. G~i), by F.7.7.2, Vi. 02 (G 1 }

Since Li < K, K is described in case (1) or (2) of 11.1.2 by 11.4.1.2. As

CH 1 (U) ::; CH 1 (V) ::; Mi and Li :::'.] Mi, we conclude from the structure of those

groups that CK(U) ::; 02,z(K). Thus we may pick an Hf-chief section W of Ug

such that F := 02 ,F· (K) is nontrivial on W. Again from the structure of K, as

V i. 02(Hf), V is nontrivial on Autp(W), so V induces a transvection on W.
But comparing the groups in 11.1.2 to those in G.6.4.2, AutH 1 (W) contains no
transvection, completing the proof of the lemma. D

By 11.5.8, K =Li, so (VG^1 ) is nonabelian by 11.5.3.3, contrary to 11.5.7. This
contradiction completes the proof of Theorem 11.0.1.