5.2. USING WEAK BN-PAIRS AND THE GREEN BOOK 653as ILi = IK11, L ~ k, contradicting M = !M(LT). This contradiction completes
the proof of the claim.
Finally we treat the case where a is the U 4 (q)-amalgam and Q ::::l k, along withthe remaining two cases where a is the amalgam of G 2 (q) or^3 D 4 (q). In these last
two cases Q is special and K is irreducible on Q/Z(Q), so as in the earlier cases of
the Ls(q) and Sp 4 (q) amalgams, there is a unique noncentral 2-chief factor under
the extension of K by a Cartan subgroup, and again we get 02 (K) = Q. Thus
in each of our three cases, Q ::::l k, so k E C(Na(Q)) by 1.2.7 ask E £*(G, T).
Further K* ~ L 2 (q) when a is the amalgam for U4(q) or G 2 (q), and K* ~ L 2 (q^3 )
when a is the^3 D 4 (q)-amalgam. As above, A.3.12 gives a proper extension with
"0 2 (B) = l" only when "B" is L2(4). This eliminates the^3 D4(q) amalgam, andforces a to be the amalgam of U 4 ( 4) or G 2 ( 4). Therefore Q ~ 41 +4 and there is X of
order 3 in ODLB(K/Q) with Q/<I!(Q) = [Q/<I!(Q), X], so X acts on k E C(Na(Q))
by 1.2.1.3. But as in our application of A.3.12 above, K* ~ A1, A1, J 1 , L 2 (25), or
L 2 (p) for p = ±1 mod 5 andp = ±3 mod 8, and X centralizes A5 ~ K* ~ K*, sowe conclude from the structure of Aut(K) that X centralizes K. Thus K* is not
A 7 , for otherwise m 3 (KX) = 3, contradicting Na(K) an SQTK-group. Further
as Q/iJ!(Q) = [Q/<I!(Q),X] is of rank 8, and 8 is not divisible by 3, K* is not A7.
Finally G. 7.2 eliminates the remaining possiblities for K. This completes the proof
of 5.2.8. D
Conclusions (1) and (2) of 5.2.8 will lead to conclusions (3) and (2) of Theo-
rem 5.2.3, respectively, so we adopt notation reflecting the groups arising in those
conclusions. Namely we define G to be ·of type Xr(q) if a is the Xr(q)-amalgam
and KE L(G,T). Define G to be of type M23 if a is the Ls(4)-amalgam and
K ¢ L*(G,T). Thus in this language, we can summarize what we have accom-
plished in 5.2.6 and 5.2.8:
THEOREM 5.2.9. One of the following holds:
(1) G is of type Ls(q), Sp4(q), G2(q),^3 D4(q), or^2 F4(q), for some even q > 2.
(2) n > 2 is even and G is of type U 4 (2nl^2 ).
(3) G is of type U5(4).
(4) G is of type M23.
5.2.2. Characterizing M 23. The remainder of this section is devoted to a
proof that:
THEOREM 5.2.10. If G is of type M23 then G is isomorphic to M23.
The proof of Theorem 5.2.10 involves a short series of reductions. Assume G is
of type M 23. Then by 5.2.8, a is the L 3 (4)-amalgam and K < k E L*(G, T) with
k an exceptional A 7 -block. Let M2 := k, M1 := M, and M1,2 := M1 n M2. Set
l/i := 02(Mi), V :=Vi, and U := l/2. Then V ~ U ~ E16 with M2/U ~ A1. Hence
we can represent M 2 /U on n = {l, ... , 7} so that T has orbits {l, 2, 3, 4}, {6, 7},
and {5} on n. Indeed:
LEMMA 5.2.11. (1) H is the global stabilizer in M2 of {6, 7}.
(2) M 1 , 2 is the global stabilizer in M2 of {5, 6, 7}.
(3) M/V ~ I'L2(4).
(4) M2 E M(T).