1150 15. THE CASE .Cf(G, T) = (/)
and K = 02 (K), we conclude using Coprime Action that K centralizes Qc. Thus
Qc = CT(TK)· By 15.3.58, U :::; Kand K* ~ L 3 (2), so K := K/U ~ L 3 (2) or
SL2(7) and hence ITKI :2: 210 and Qc:::; if.?(TK)·
Next X = [X, 02 (NK(V 0 ))], so as X n QH = E:::; U by 15.3.56.3, X:::; K.
Thus as TK = J(j[tQc and Q 0 :::; if.?(TK), TK = (X,Xt)u. Further X:::; (UY)=
YU, so TK :::; YU. Then (2) holds as ITKI :2: 210 and IYUl2 = 210 by 15.3.54.4
since s = 1.
Since F*(YU) = 02 (YU), since V = Z(P) by 15.3.55, and since Qc centralizes
TK, QcV = CyuQ 0 (P). Thus by Coprime Action, QcV = Qy x V, where Qy :=
CQ 0 v(Y). Then as Tacts on Qy, and D 1 (Z(T)) = Z ~ V, Qy = 1, so Qc ~ Z,
establishing (1) and (3). Then 02 (YT) = 02 (Y), so as Y :'.SJ M, F*(M) =
02 (M) = 02 (Y) using A.1.6. By 15.3.58.2, s = 1, so from 15.3.53.1 and 15.3.54.4,
Auty(B) = 02 (NaL(B)(Auty(B))) for BE {V,0 2 (Y)/V}. Therefore Y = 02 (M)
by Coprime Action, so ( 4) holds. D
Let DM E Syl 3 (CM(V 1 )) and DH E Sy[s(H); observe DM and DH both have
order 3. Let (v) =Zs n Vi and Z = (z). By 15.3.46.5, Ca(v):::; M.
By 15.3.59.4, M =YT, ands= 1 by 15.3.58.2, so by 15.3.54.4, CM(DM) =
DM x JM, where JM ~ 84 and V1 = 02(JM)· By construction, an involution
t E JM - Vi induces a transvection on V, and hence t ¢. UCT(V).
Next a Sylow 2-group of CM(DM) is dihedral of order 8 with center (v), and asCa(v) :::; M, ICa(DM )b = 8. On the other hand, from the structure of H described
in 15.3.58and15.3.59, ICH(DH)l2 = 24 , so DM ¢.Di}. Thus as DH E Syls(Ca(z)),
t ¢. zG. Summarizing:
LEMMA 15.3.60. (1) DM ¢_ D<fr.
(2) An involution t in T n JM - V 1 is not in UCT(V), and t fj_ z^0.
LEMMA 15.3.61. (1) t ¢_VG.
(2) All involutions in K are in zG or v^0.
PROOF. As U = U 1 EB Uf by 15.3.58.3, m(U) = 2m([U, t]), so U is transitive
on involutions in iU. Thus 02 (CH·(t)) = 02 (CH(t)), and hence t centralizes a
conjugate of DH. But by 15.3.46.5, CM(v) = Ca(v), so DM is Sylow in Ca(v) by
construction. Thus (1) follows from 15.3.60.1.
From the action of Hon U described in 15.3.58, H has two orbits on involutions
in U - Z: (U1 - Z) U (Uf-Z) ~ z^0 and vH. Let a EV - U with Ua the preimage
in U of Cu(a). Then all involutions in K - U are fused into aUa under H, so it
remains to·show that each such involution is in z^0 U v^0. Now IU : Ual = 4 = IUI
by 15.3.53, so Ua = Cu(V) = Cu(U n V). Thus Ua S:! E 4 x Ds, and all involutions
in Ua V are in the two E 16 subgroups A 1 and A 2 of Ua V.
Next VUa :::; P; let p+ := P/V. From the description of I in 15.3.54, U;i: =
[P+, U] is an isotropic line in the orthogonal space p+ with one singular point, and
I is transitive on singular and nonsingular points of p+. Thus At, i = 1, 2, are the
nonsingular points in U;i:. Therefore there is Di of order 3 in I centralizing At and
[Z, Di] is a singular line in the orthogonal space V, so [Di, Z] :::; V..L = UnV. Let ai
generate CAJDi). If ai EU, then each member of Ai is fused into U under Di, so
that (2) holds. Thus we may assume ai ¢. U. Here each member of Ai - (ai)[ai, P]
is fused into U, and P is transitive on ai [ ai, P], so it remains to show the ai is fused
to z or v.