936 13. MID-SIZE GROUPS OVER F2
LEMMA 13.7.3. (1) VH :S QH.
(2) UH :S Z(VH), so that VH :S Ha.
(3) (Ufi) :S 02(LT) = Q.
(4) For h E H, either [V, Vh] = 1, or [V, Vh] = [V, VH] = Vi with VH = fTh =
((5,6)).
( 5) Either V H is abelian, or <l? (V H) = Vi.
(6} 02(L1) :S QH :S R1, Vi= [V, QH], Vi= [Vi, QH] =[UH, QH] = Cvs(QH),and [VH,QH] =UH·
(7) Either
(i} Ho :SQ, so Ho= CH(VH), or
(ii} IHo : Q n Hol = 2, so Ho= ((5,6)), [VH,Hol =Vi, and Ho :S
CHCVH).
(8) If L/02(L) ~ A.6, then VH is abelian.
(9) H n M = NH(V) and L 1 = ()(H n M).
PROOF. As UH is abelian,UH :S Cr(Vi) = CrCV),
so V :S CH(UH) = QH and hence (1) holds. By (1) and F.9.3, V:::; CQH(UH), so
(2) and (3) hold.
Let h EH. Then by (2),
v h :S CQH(Vi) = CQH(V), -
so (4) holds, since (5, 6) is the transvection inf' with center Vi. Then (4) implies
(5).
If [L1, QH] :S Q = CT(V), then [L1, QH] :::; CL 1 (Vi); so as Li/CL 1 (Vi) ~ A4has trivial centralizer in GL(Vi), [Vi, QH] = 1, contrary to 13.5.4.5 since 02(G1) :S
QH. Thus [L1,QH] i. Q, so 02(L1) = [QH,L1] :::; QH :S R1, and hence Vi =
[V, 02(L1)] = [V, QH]· Then as UH= (Vl), (6) holds.
Observe Ho:::; C.R 1 (Vi) = 1 or ((5, 6)). If Ho= 1, then (7i) holds. Otherwise
Ho= ((5, 6)), and then as [V, (5, 6)] = V1, [V, Ho]= V1, so that (7ii) holds.
If L/0 2 (L) ~ A 6 , then each t E T inducing a transposition on L invertsLo/02(Lo) (see Notation 13.2.1), and hence t tj. QH as Lo :S L1:::; H. We conclude
[V, Vh] = 1 for all h EH-since if not, some t E Vh induces a transposition on L
by (4), whereas Vh :S Q:fl by (1), contrary tot fl. QH. Thus (8) is established.
Finally as H:::; Gz, H n M = NH(V) by 12.2.6, so the remaining statement of
(9) follows using 13.3.7. DBy 13.7.2, UH :S Ho.
LEMMA 13.7.4. (1) If L/02(L) ~ A6, then H* is faithful on UH/CuH(QH)·
(2) There is an H-isomorphism cp from QH/Ho to the dual of UH/CuH(QH),
defined by cp(xHo): uCuH(QH) f--7 [x, u].
PROOF. Part (2) holds by F.9.7, so it remains to establish (1). Set U 0 :=
Cu~(QH) and observe QH = CH(UH) :SC:= CH(UH/Uo). On the other hand
V3 = [%, L 1 ] with V1 = V n U 0 by 13.7.3.6, so that L 1 i. C.