S14 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALSFurth~rmore HE He by 1.1.4.6. So as K f:. Mand R acts on K, we have the
hypotheses of C.2.7, and we conclude K appears on the list of C.2.7.3. In 1.3.4.2,
K/0 2 (K) f'::! (S)L 3 (p), whereas no such K appears in C.2.7.3. So we have reduced
to 1.3.4.1 where K/0 2 (K) f'::! L 2 (2n) or L 2 (q) for q an odd prime. Then by C.2.7.3,
either K is an L 2 (2n)-block or an As-block, or else K/02(K) f'::! Lz(7) f'::! L3(2).
But the latter two cases are eliminated as PT = T P with p > 3. Therefore K is
an L 2 (2n)-block and 1.3.4.1 holds, so H = KKtT, where t E NT(P) - NT(K). In
particular [K, Kt] = 1 as distinct blocks commute by C.1.9, so X = WWt withW := 02 (X n K) and [W, Wt] = 1. Thus
X/Z(X) = WZ(X)/Z(X) x wt Z(X)/Z(X)
and then Na(X) permutes {WZ(X), wtz(X)} by the Krull-Schmidt Theorem
A.1.15. In particular, L = 02 (L) acts on WZ(X). This is impossible, as NL(P) is
irreducible on P/i!J(P) from the structure of Lin cases (c) and (d) of 1.2.1.4.
The proof of 1.3.8 is complete. DAs in the previous section, we want to study the action of members of our new
class of solvable uniqueness subgroups on their internal modules. So let 'Bt(G,T)
consist of those XE S(G, T) with XE Xf, and let 'Bj(G, T) := 'Bt(G, T)n'B*(G, T).LEMMA 1.3.9. Let XE S(G, T), LE .C(G, T), and X:::; K :=(LT). Then
(1) If L/02(L) is quasisimple, then LE C*(G,T).
(2) If XE 'Bt(G, T), then V(X, CTnK(X/02(X))):::; V(K) and LE Ct(G, T).PROOF. Part (2) follows from A.4.10, just as in the proof of 1.2.9. Thus it
remains to establish ( 1).
Assume L/02(L) is quasisimple. Then Lis described in 1.3.4. In cases (2)-
(4) of 1.3.4, L E .C*(G, T) by 1.2.8.4-unless possibly L/0 2 (L) f'::! L 4 (2). But inthe latter case, if (1) fails, then L < Y E .C(G, T), and from 1.2.4 and A.3.12,
Y/02(Y) f'::! Ls(2), M 2 4, or J4. Now X = 02 (X)P with P f'::! E 9 and NT(P) is
irreducible on P, so Tacts nontrivially on the Dynkin diagram of L/0 2 (L) f'::! L 4 (2).
This is impossible, as no such outer automorphism is induced in Aut(Y/0 2 (Y)).
Therefore 1.3.4.1 must hold. Then by 1.2.8.2, P f'::! E 9 , L/0 2 (L) f'::! As, and
Y/02(Y) f'::! Ji or Lz(p). But again as NT(P) is irreducible on P, some element
of NT(L) induces an outer automorphism on L/0 2 (L) f'::! As, whereas no such
automorphism is induced in NT(Y).
Thus 1.3.9 is established. D1.4. Properties of some uniqueness subgroups
In this section we summarize some basic properties of the families C * ( G, T)
and S* ( G, T) of uniqueness subgroups, which will be used heavily later.So we consider some L contained either in .C(G, T) or in S(G, T). Note that
the assertion in 1.4.1.1 below is the starting point (as we just saw in the proof of
1.3.8) for arguments using pushing up (sections C.2 etc.).
LEMMA 1.4.1. Let L E C(G, T) U S(G, T) and set Lo := (LT) and Q :=
02(LoT). Then M := Na(Lo) =!M(LoT), so LoT and Na(Q) are both uniqueness
subgroups, and