14.3. FIRST STEPS; REDUCING (vG1) NONABELIAN TO EXTRASPECIAL 989Thus it remains to deal with the case where a is the^2 F 4 (2)-amalgam or the
Tits amalgam. The subgroups G1 and G2 are described in section 3 of [Asc82b].
In particular E := [QH, QH] ~ E32, and Z(QH) = F, where F := CH(E). FurtherF = E if a the Tits amalgam, while if a is the^2 F 4 (2) amalgam, then F = (v 5 )E
with (v5) := CQH(K) ~ Z4. In particular F .:::; Q by (*). Next His irreducible
on QH/F of rank 4, so Q =For QH. In the former case, F and E = D 1 (F) arenormal in M; in the latter, E = [Q, Q] and F = CQ(E) are normal in M.
Now H/F:::; Mc/F, with Mc/F contained in the stabilizer A~ L 4 (2)/E 16 of
Zin GL(E), and H/F ~ Sz(2)/E 16 or D10/E 16 contains a Sylow 2-group T/F of
Mc/F, with QH/F = 02(A). Thus QH ::::) Mc, so Q = QH using(*). Further
the Sylow 2-group T/Q of Mc/Q is cyclic, so by Cyclic Sylow-2 Subgroups A.1.38,
Mc/Q is 2-nilpotent. Therefore Kc/Q = O(Mc/Q) is of odd order and contains
K/Q ~ Z5; then as K < Kc, Kc/Q ~ Z15 from the structure of L4(2). But by
3.2.11 in [Asc82b], H is transitive on the involutions in Q - F, so if j is such an
involution, then Mc =HG Mc (j) by a Frattini Argument. In particular, j centralizes
an element of order 3 in Mc, impossible as Kc/Q of order 15 is regular on (Q/F)#.
This completes the proof of 14.2.21. D
By 14.2.21, a is isomorphic to the amalgam of G, where G is^2 F4(2), the Tits
group^2 F4(2)', G2(2), G2(2)' ~ U3(3), M12, or Aut(M12). As G and G are both
faithful completions of the amalgam a, there exist injections f3J : G J -+ G J of the
parabolics G J, G J for each 0 i= J ~ {1, 2}, such that /31,2 is the restriction of f3i to
G 1 , 2 and f3i( Gi) = Gi for i = 1, 2. We abuse notation and write f3 for each of the
maps f3J. Let T := 13-^1 (T).
LEMMA 14.2.22. (1) a is not the amalgam of^2 F4(2), G2(2), or Aut(M12).
(2) If a is the amalgam of G2(2)', M12, or^2 F4(2)', then G ~ G.
PROOF. First if a is of type^2 F4(2)',^2 F4(2), or G2(2), then G1 = H =Mc=Ca(Z) by 14.2.21, so that the hypotheses of Theorem F.4.31 are satisfied. Then
G ~ G by F.4.31, and hence as G is simple, a is the amalgam of^2 F 4 (2)' and
G ~^2 F 4 (2)', so that (2) holds.
Thus we may assume that a is of type G2(2)', M12, or Aut(M12).
Suppose first that a rs of type Aut(M12). Let R := f3(T n 02 (0)). Then
J(T) ~ E 16 is normal in YT and M =YT by 14.2.21, so M controls fusion in J(T)
by Burnside's Fusion Lemma A.1.35. Thus for j E J(T) - R, ja n J(T) n R = 0.
But each involution in R is fused into J(T) n R under G1 U G2, so ja n R = 0, and
hence j tJ_ 02 (G) by Thompson Transfer, contrary to the simplicity of G.
In the remaining cases we appeal to existing classification theorems stated in
Volume I: If a is of type M12, then G ~ M12 by I.4.6, and if a is of type G2(2)',
then G ~ G2(2)' by I.4.4. D
Notice 14.2.21 and 14.2.22 establish Theorem 14.2.20.14.3. First steps; reducing (VG^1 ) nonabelian to extraspecial
As mentioned at the beginning of the chapter, the work of the previous two
sections allows us to treat the most important subcase of the case £1(G, T) = 0
where M1/CMt(V(M1)) ~ L 2 (2) in parallel with the final case L/02(L) ~ L 3 (2)
in the Fundamental Setup (3.2.1). As usual we define an appropriate hypothesis,
which excludes the quasithin examples characterized in earlier sections.