2.4. THE CASE WHERE rg IS NONEMPTY 5292.4.1. Shadows of groups of rank 2 with L 2 (2n)-blocks. In this subsec-
tion we continue the proof of Theorem 2.4.1 by eliminating the shadows of L 3 (2n)
and Sp4(2n) extended by an outer automophism nontrivial on the Dynkin diagram.
To be more precise, we will show that if Lis an L 2 (2n)-block, then H essentially
has the structure of a maximal parabolic of L 3 (2n) or Sp 4 (2n). Then we will show
that 02 (G) < G via transfer.
The main result of this subsection is:
THEOREM 2.4.7. L is not an L 2 (2n)-block for n > 1.
Throughout this subsection, G and H continue to be a counterexample to
Theorem 2.4.1, with H = L 0 S and Lo= (L^8 ). Moreover we also assume that His
a counterexample to Theorem 2.4.7, so that Lis an L 2 (2n)-block with n > 1. Set
q := 2n. Fix a Hall 2'-subgroup D of Lon M normalizing Lon S; thus Lon M =
(Lon S)D is a Borel subgroup of Lo. Of course D -I 1 as n > 1.
The proof divides into two cases: s = 1 and s = 2. Further the case where
s = 1 is by far the more difficult, as that is where the shadows of L3(q) and Sp4(q)
extended by outer automorphisms arise. Thus the treatment of that case involves
a long series of lemmas.
In the remainder of this subsection, set
R := J(S).
The Case s = 1.
Until this case is complete, we assume that s = 1, so that Lo = Li = L, and
H=LS.
LEMMA 2.4.8. (1) IT: SI = 2. Hence T normalizes S and R.
{2) 02(L) = U = Q = 02(H).
(3) R = Baum(S) = uux E Sylz(L), Un ux = Z(R), and A(S) = A(T) =
{U, ux} for each x ET - S.
( 4) Dx acts on L, and either
(a) Z(L) = 1 and L ~ P^00 for P a maximal parabolic in L3(q), or
{b) Z(L) ~ E 2 n, Dx is regular on Z(L)#, and L ~ P^00 for P a maximal
parabolic in Sp4(q).
(5) (T, D) = TB, where B is an abelian Hall 21 -subgroup of (T, D) containing
D, S normalizes RD, R :::) RB :::) BT, CR(B) = 1, and Tis the split extension
of R by NT(B). If x E NT(B) - S, then B = DDx.
(6) If Z(L) -=f. 1, then B = D x Dx; while if Z(L) = 1, then AutB(Z(R)) =
Autn(Z(R)) ~ D is regular on Z(R)#. ·
(7) U and ux are the maximal elementary abelian subgroups of R.
PROOF. By 2:4.5.3, we have the hypotheses of Theorem C.6.1, with T, S in
the roles of "A, TH", so we may appeal to Theorem C.6.1. In particular, conclusion
(a) of C.6.1.6 holds, since L is of type L 2 (2n) for n > 1. Thus (1) holds and
R = J(S) = J(T). By C.6.1.1, Baum(S) = R = QQx for each x E T - S and by
C.6.1.3, {Q, Qx} = A(S). As R = QQx with Q E A(S), Q n Qx = Z(R) using
B.2.3.7. Since Qx f:. Q, Qx is an FF-offender on U by Thompson Factorization
B.2.15, so as H =LS with Lan L 2 (2n)-block, R/Q = QQx /Q is Sylow in LQ/Q