i4.4. FINISHING THE TREATMENT OF (vG1) NONABELIAN ioo7Conclusion (10) of G.11.2 is impossible, as m3(H*) = m 3 (H):::; 2 since His an
SQTK-group. As LiT :::; H with LiT /02(LiT) ~ 83, conclusion (11) of G.11.2
does not hold. Conclusions (8) and (9) of G.11.2 appear as conclusion (4) and (5)of 14.4.2. So it remains to show that conclusion (7) of G.11.2 leads to conclusion
(3) of 14.4.2. In case (7) of G.11.2, W^9 * i K*. Then as we saw W9* :SI CH* ("V 2 ),
it follows that W9* = (x*)Y*, where x* is an involution interchanging the twocomponents of H, m([U, x]) = 4, and Y = Cr•nK• (x), as desired. D
14.4.1. Characterizing G 2 (3) when d = 4. The only quasithin example
satisfying Hypotheses 14.3.1 with U extraspecial is G2(3), occurring when d = 4,
so our first main result treats this case:
THEOREM 14.4.3. Assume Hypothesis 14.3.10 with U extraspecial. If d = 4,
then G ~ G2(3).Until the proof of Theorem 14.4.3 is complete, assume G is a counterexample.
LEMMA 14.4.4. {1) K* ~ Eg.{2) V = E and W^9 is of order 2, inverts K, and is generated by an involution
of type c2 on U.
{3) Either H = KW^9 , or H ~ 83 x 83.
(4) Lis an L 3 (2)-block with V = 02(L).
(5) UW9 E 8yb(L) and U = 02(Li).
{6) L does not split over V, and m 2 (UW9) = 3.
(7) H =KT and M =LT.
PROOF. As d = 4, case (1) or (2) of 14.4.2 holds, establishing (1) and (3) since
m(W^9 *) = 1 by 14.4.2. By 14.3.23.1, V = E. Thus the first two statements in (2)
are established. By (1), H is a {2, 3}-group. As Li :SI H* by (1), U = [U, Li] by12.8.5.1, so that U = [U,Li] :::; L. By 12.8.8.4, 02(LU) = 02(L) is described in
G.2.5. Therefore since E = V and m(U/V) = 2 = m 2 (02(Li)), V = 02 (L), giving
(4); and [J = 02 (Li) so U = 02 (Li), and hence TL:= TnL = UW^9 , giving (5).
Let a E W9 - U. Then a inverts Li with U = [U, Li], so using the structure
of ot(2), either the remaining two statements of (2) hold, or a* is of type a2,
A := (a, [U,a]) ~ E16, and H = NH(A)Li. In the latter case, a acts on a
complement to Vin U, so that UW9 splits over V; then by Gaschiitz's Theorem
A.1.39, L splits over V. Conversely if L splits over V, then from the structure of
the split extension of Es by L 3 (2), J(TL) ~ El6, so a* is of.type a 2 and A= J(TL)·
Thus to complete the proof of (2) and (6), it remains to assume L splits over V,
and obtain a contradiction. Set N+ := Na(A)/Ca(A); then [02(L2), L2] =A, so
that LtTt ~ Z2 x 83 while N.K(A)+ ~ A4. Then from the structure of Aut(A) ~
GL4(2), the subgroup of N+ generated by L2T and N K(A) is isomorphic to A 7. But
then the stabilizer of z in this subgroup is L 3 (2), contradicting Ha {2, 3}-group.
For (7), observe V = 02(L) :SI M by (4). Then as H =KT and KQ/Q ~ E 9
by (1), H n M = LiT, so that M =LT by 14.3.7. D
LEMMA 14.4.5. (1) L = M.
{2) U = 02(H) and H* ~ Z2/Eg.
(3) T = UW^9 •
PROOF. Let Ki := (WgH). By (1) and (2) of 14.4.4, Ki is K* ~ E 9 extended