i3.S. THE TREATMENT OF As AND Aa WHEN (v;^1 ) IS NONABELIAN. 9i9
In the remainder of the section, H will denote any member of 'Hz with UH
nonabelian.
Then <P(UH) = Vi by F.9.2.2. By F.9.4.1, V i. QH, while by F.9.2.1, Vi :::; QH.
Thus as IV: Vil= 2, V3 = V n QH and V* is of order 2. By F.9.2.1, UH E R 2 (H).
LEMMA 13.5.13. (1) If g EH with Vi< V n VB, then (V, VB) is a 2-group.(2) The hypotheses of F.9.5.5 and F.9.5.6 hold.
PROOF. As OH(UH) = QH is a 2-group, we may assume V* -j. VB*; so as V*
is of order 2, V n VB :::; V n QH =Vi. Then as Li is transitive on f!;t, we may
take Vi :::; V n VB. Now 13.5.11 and 13.5.6.3 show that VB normalizes V, and so
(1) follows.
By (1), we have the hypothesis ofF.9.5.5. Further by 13.5.5, OH(Vi) :::; OM(Vi).
Now if n = 5 then OM(Vi) = OM(V), while if n = 6 then OM(Vi) is trivial orinduces transvections with center Vi on V. Thus we also have the hypotheses for
F.9.5.6. D
LEMMA 13.5.14. If n = 6, assume UH = [UH, Li]. Let l E L - LiT, and
if n = 6, choose l to fix a point w E n fixed by Li. Set K .-(UH, UJ,.) and. L_ := 02 (0 2 (LT)K). Then
(1) If UH= [UH,Li] then UH= [UH, Li]:::; Li:::; L.
(2) UH= 02(Li) ~ E4.
(3) If n = 5 then K =Land L = L_, while if n = 6, then K ~As is the
stabilizer in L of w. Thus in any case Li :::; K.
(4) The hypotheses of G.2.4 are satisfied with Vi, Vi, V, L_, UH, K in the
roles of "Vi, V, VL, L, U, I", so K = L_ and K is described in that lemma.PROOF. Suppose first that UH = [UH, Li]. Then as Vi :::; [Vi, Li], UH
[UH, Li]. Thus (1) holds. Moreover if n = 6, then UH = [UH, Li] by hypothesis, so
UH:::; L by (1).As UH= (VP) is nonabelian, UH "I-1, and as LiT:::; H:::; NH(UH ), UH :s:l Lit'.
Thus (2) holds if n = 5. Similarly if n = 6, then UH :::; L by the first paragraph, so
as UH :s:l Lit', (2) holds again. Part (3) is immediate from (2) and the choice of l.Then (3) implies the first statement in (4). Finally Li :::; 02 (K) = L_ by (3) and
UH:::; Li by (1), so that K = LUH = L by G.2.4. D
13.5.2.1. Identifying the groups. In the branch of the argument that will lead
to the groups in Theorem 13.5.12, Li :s:l Gi and Gi is the unique member of 'Hz.
We begin by deriving some elementary consequences of the hypothesis that Li is
normal in some member H of 'Hz with UH nonabelian.LEMMA 13.5.15. Assume Li. :s:l H, UH is nonabelian, and L/0 2 (L) is not A5.
Then(1) UH= [UH, Li]:::; L and Li~ Z3.
(2) UH= 02(Li) ~ E4.
Choose l and K :=(UH, UJr) as in 13.5.14. Then
(3) K is an As-block contained in L.
(4) If n = 5 then L = K, so that L is an As-block; if n = 6, then L is an
A5-block.