1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

590 3. DETERMINING THE CASES FOR LE .Cj(G, T)

an odd prime p. Here T'K is dihedral of order greater than 4 and self-centralizing

in Aut(K), so that NH·(T'K) is a 2-group, and then D = 1, contrary to(*).

Thus case (9) of A.3.12 holds, with K ~ L3(p). If Y ~ 8L2(p), Y*

OK·(Z(T'K))^00 is D-invariant, again contrary to().

In the remaining subcase of (9), K ~ U3(5), with Y S=! A5. Here X* =

02 (0Aut(K•)(T'K)) is of order 3 and induces outer automorphisms on K* with

OK·(X*) the double covering of 85 which is not GL2(5). We conclude D* = X*.

Finally K/0 2 (K) is not 8U 3 (5) by A.3.18. Therefore K/02(K) ~ U3(5), so case

(c) of conclusion (3) holds..

This completes the treatment of the cases appearing in A.3.12, and hence com-

pletes the proof of the lemma. D

LEMMA 3.3.13. If HE H(T) with H/0 2 (H) ~ 83 wr Z2, then D :S Na(H).

PROOF. Let Ho := (H, D); by 3.3.10.1, 02(Ho) -:/=-1. Set Y := 02 (H) and

notice Y E 3(G, T). If D normalizes Y, then D normalizes YT = H and the

lemma holds, so we assume that D does not act on Y. Therefore Y is not normal

in H 0 , so by 1.3.4, Y < K 0 := (KT) for some K E C(H 0 ), and Ko is a normal

subgroup of Ho described in cases (1)-(4) of 1.3.4 with 3 in the role of "p". Let

(KoTD)* := KoTD /OKoTD(Ko/02(Ko)). Notice that 02(Ko) :S 02(Ho) :S 02(H)

using A.1.6, so that ND·(Y) = ND(Y). Hence D does not act on Y and in

particular D* -:/=-1, so that

(*) T* is not self-normalizing in K*T* D*.

Further H* /02(H*) ~ H/02(H) ~ ot(2), so

(*) T /02(YT) ~ Ds.

Inspecting the list in 1.3.4 for cases in which (*) and (**) are satisfied, we

conclude that case (1) of 1.3.4 holds, with K* ~ L 2 (2n) for 2n = 1 mod 3, and H*

is contained in the T-invariant Borel subgroup B* of Ki). As D* acts on T*, D*

acts on B and hence also on the characteristic subgroup Y of B*, contrary to an

earlier remark. This completes the proof. D

LEMMA 3.3.14. L = M+ ~ M, eliminating case (1) of 3.3.8.

PROOF. Assume otherwise. Then by 3.3.10.3, Li~ L 3 (2). Therefore M+T =

(Hi, H2), where Hi := (Hi,i, T) and fli,i, i = 1, 2, are the maximal parabolics of

Li over T n Li. Notice that Hi/02(Hi) ~ 83 wr Z2, so by 3.3.13, D normalizes

Hi. But then D normalizes M+T = (Hi, H 2 ), contrary to 3.3.6.b. D

Our next lemma puts us in a position to exploit an argument much like that

in the proof of 3.3.14, to eliminate many cases where Lis generated by a pair of
members of C(L, T).

LEMMA 3.3.15. Suppose LT = (Yi, Y 2 , T) with 1j E C(L, T). Set Hj :=

(lj,TD), and assume 02(Hj) f=-1forj=1and2. Thenfori = 1 or2: D does not

normalize Yi, Yi/02(Yi) ~ L2(4) or A5, Yi <KE C(Hi) such that K/0 2 (K) ~Ji

or U3(5), respectively, K ~ Hi, and D i M. When K/0 2 (K) ~ Ji, T induces

inner automorphisms on Yi/02(Yi) and Kn Di M.

PROOF. Notice lj, Hj satisfy the hypotheses of 3.3.12 in the roles of "Y, H",

so we can appeal to that lemma. Suppose D normalizes both Yi and Y 2. Then

D normalizes (Yi, Y2, T) =LT, contradicting 3.3.6.b. Thus D does not normalize
some Yi, so the pair Yi, Hi is described in case (b) or ( c) of 3.3.12.3. Further D i M
by 3.3.6.a, and when K/02(K) ~Ji, Kn Di M by 3.3.12. D