1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

960 13. MID-SIZE GROUPS OVER F2

Thus m(U H) = 6. Since the case with a single nontrivial 2-chief factor which is an

A 8 -module is excluded by 13.7.6.3, [W,K] #-1, so we can appeal to 13.8.22.

If u; contains a strong FF* -offender on u H' then B.3.2.6 says that u; ~ E16

is generated by the transpositions in T* and m(UH/CuH(U 1 )) = 3. Thus as u; E

Q(H*, UH),

m(W /Cw(U 1 )) s 2m(U;) - 3 = 5;

so as W is a faithful module for H* ~ 88 , we conclude [W, HJ is the 6-dimensional

module or its 7-dimensional cover, and m(W/Cw(U 1 )) = 3. Thus

m(UH/CuH(U;);:::: m(UH/C(JH(U;)) +m(W/Cw(u;)) = 6.

Now by 13.8.8, m(UH/DH) s m(U;) = 4, sou; does not centralize DH· Then

by F.9.13.6, Ai= [DH, u,] s UH. So as the transpositions in u; induce transvec-

tions on UH with distinct centers, we conclude Cu;(DH) is a hyperplane of u;, so

m(UH/CuH(U 1 )) S 5, contrary to our previous calculation.

Therefore u; contains no strong FF* -offender on u H' so by 13.8.22 either

(i) u; ~ Z2 induces a transvection with center Ai on w, or a transvection

with center A1 on UH, or

(ii) Ai i W, and u; contains a strong FF*-offender on W.

In case (ii), as in the previous paragraph, we conclude u; ~ E15 and W is the
orthogonal module, leading to the same contradiction.
So case (i) holds. Then m(U;) = 1 and [W, K] i=-1, so 13.8.23.4 supplies a

contradiction, since CH· ( CE(h)) = (h) for each faithful F 2 H* -module E (namely

with [E, K] of dimension 6 or 7) on which some h* E H* induces a transvection.

This contradiction completes the elimination of the case L/0 2 (L) ~ A 6 •

Therefore L/0 2 (L) ~ A 6. We eliminated n = 3 earlier, so n = 4 or 5. As T

acts on the two minmal parabolics determined by Lo and Li,+, TK: = T*. Further

as Li :SI H n M, LiT = H n M. Observe that LiTK: is a parabolic of rank 2

determined by two nodes not adjacent in the Dynkin diagram.

Suppose n = 5. By 13.2.2.9, NK(L 0 ) s Kn M s NK(Li,+), the node f3

determined by L 0 is an interior node, and the node o determined by Li,+ is the

unique node not adjacent to (3. Thus we may take o and f3 to be the first and third

nodes of the diagram for H*. Then LiT S H 2 SH with H 2 /0 2 (H 2 ) ~ 83 x L3(2).

As LiT = H n M, H2 i M, so H2 E 1-lz, contrary to 13.8.21.1.

Therefore we have established that n = 4 in each case of 13.8.8. As mentioned

earlier, we can now show that (2) holds: For H s Gi, so K s Ki E C(G1)
by 1.2.4. Now KiT E 1-lz and we have shown Ki/02(K1) is not L5(2). Hence
Ki =KE C(Gi) by 13.8.21.2 and A.3.12. As G 1 E 1-lz, we conclude from 13.8.21.3
that H =KT= KiT = Gi, so that (2) holds.
Thus (2) is established, and we've shown that L/0 2 (L) ~ A 6 and H* ~ L4(2).

We may assume (1) fails, so that UH is not the natural module for H*. Now

u; E Q(H*,UH), so UH is of dimension 4 or 6 by B.4.2 and B.4.5. Then as

the maximal parabolic LiT* determined by the end nodes stabilizes the line V 3 ,

dim(UH) = 4. As (1) fails, [W, K] i=-1; hence we can appeal to 13.8.22 and 13.8.23.

Moreover m(V,;) > 1by13.8.18.4.