982 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L'.f(G, T) IS EMPTY(8) For each HE H*(T, M), H n M is the unique maximal subgroup contain-
ing T, and H is a minimal parabolic described in B.6.8, and in E.2.2 when H is
nonsolvable.PROOF. Parts (1)-(6) follow from 14.1.18, so it remains to prove (7) and (8).
As Z = D 1 (Z(T)) is of order 2, Na(T) :::; Ca(Z) =Mc. As Na(T) pre~erves :S,
Na(T) :::; M by 14.2.1.3, completing the proof of (7). Then (8) follows from (7)
just as in the proof of 3.3.2.4. D
For the remainder of the section, H will denote a member of H(T, M).
Set MH := Mn H, UH := (VH), QH := 02(H), and H* := H/QH. Let
Mc:= Mc/Z. Since T:::;, Hi_ M, we conclude from 14.2.2.5 that:
LEMMA 14.2.3. Ca(Z) =Mc= !M(H).
In particular fr:= H/Z makes sense. Next observe using 14.2.2 that:LEMMA 14.2.4. Case (2) of Hypothesis 12.8.1 is satisfied with Y in the role of
"L".
In Notation 12.8.2, we have V2 = V, L 2 ~ Y, Vi = Z, and Li = 1. Defining
Hz as in Notation 12.8.2, 14.2.3 says:
LEMMA 14.2.5. Hz= H(T, M).By 14.2.5, results from section 12.8 apply to H. In particular recall from 12.8.4
that:
LEMMA 14.2.6. (1) Hypothesis G.2.1 is satisfied.(2) (JH:::; n1(Z(QH)) and (JH E R2(H).
(3) <!!(UH) :S V1.
(4) QH = CH(UH)·
Part (2) of Hypothesis 14.2.l excludes the quasithin examples L3(2) and A5,
which will be treated in the final section of the next chapter. In the remainder of this
section, we will identify the other quasithin examples corresponding to L ~ L 2 (2),
which do satisfy Hypothesis 14.2.1. These examples arise in the cases where some
HE H*(T, M) has one of three possible structures: n(H) > 1; H/0 2 (H) ~ D10 or
Sz(2) ~ F 20 ; or H/0 2 (H) ~ L2(2). In each case we will show that G possesses a
weak EN-pair of rank 2, as discussed in section F.1; then we appeal to section F.l
and the subsequent sections in chapter F of Volume I, to identify G. Then in latersections we show that no further quasithin groups arise under Hypothesis 14.2.1,
although certain sha?-ows are eliminated in those sections.is:14.2.1. The treatment of n(H) > 1. The first major result of this section
THEOREM 14.2.7. Either
(1) n(H) = 1 for each HE H*(T,M), or(2) G is^3 D4(2), J2, or J3.
Until the proof of Theorem 14.2.7 is complete, we assume HE H*(T, M) with
n(H) > 1. By 14.2.2.8 and E.1.13, the structure of H is described in E.2.2. As