15.3. THE ELIMINATION OF Mr/CMr(V(Mr)) =Sa wr Z 2 1147PROOF. First U is nonabelian by 15.3.52, so that Z = cl>(U) by 15.3.48.2; hence
U = UoZu, where Uo is extraspecial and Zu := Z(U). Thus we must show that
Zu = Z. As U = (Zff) is nonabelian and Zs is of order 4, Zs n Zu = Z; then as
Cv-(T) = Zs, V n Zu = Z. Therefore [V, Zu] ::; V n Zu = Z, but no member of
M induces a transvection on V with singular center, so Zu :$ Cu(V) = W. Hence
also Zij ::; Wg.
As Zu n V = z, Zij n V = Z9, so by 15.3.54.3,
zg nu= zg n (u n U^9 ) = zg n (z.i n zg.L) ::; v n z.i = LThen as Wg normalizes U and W normalizes U9 by 15.3.54.5, [Zij, W] ::; Zij n U =
1, so as P = WW9 by 15.3.54.2, Zij ·:S Zp := Z(P). Therefore also Zu ::; Zp.
By 15.3.54.4, the irreducibles for Yon P/V are not isomorphic to those on V, so
Zp = V EB Zo, where Zo is the sum of the Y-irreducibles on Zp not isomorphic
to those on V. Thus T acts on Zo, so as Z ::; V, Z 0 = 1. Thus Zp = V, so as
Zu::; Zp, Zu = V n Zu = Z, completing the proof. D
LEMMA 15.3.56. Let y E Y1 - 02 (Y1)' Vo := (zYi)' F := u n HY' x := FY'
E := F n FY, and t ET-UCr(V). Then
(1) The power map and commutator map make U into an orthogonal space withH* ::; O(U).
(2) m(U) = 2(s + 2).
(3) X n QH = E, [X,F] :SE, Vo= ZZY, and Eis totally singular of rank
s + 2 in the orthogonal space U.
(4) X* 9;! E 2 s+i induces the full group of tranvections on E with center V 0 •
(5) U = E EB JJ;t and X* ind'l),ces the full group of transvections on jj;t with axis
CEt -------(Vo).
(6) X n Xt = V is of order 2, and X X*t 9;! D8.
(1) Zs =Cu-( (X, t)).
PROOF. Part (1) follows from 15.3.55. By 15.3.54.4, JP[ = 248 +4, while by
parts (2) and (3) of 15.3.54, JPI = 22 (m(W)-l). Thus m(W) = 2s + 3. By 15.3.53.1
and 15.3.54.1, m(U /W) = 2, so (2) follows.
As y E Y1 - 02(Y1), zY E z.i - Z, so zY E U - Z by 15.3.53.2. Thus as
U is extraspecial, JU : FJ = 2; and the argument in 8.14 of [Asc94], which is
essentially repeated in the proof of G.2.3, gives us the structure of J := (U, UY):
J/02(J) 9;! 83, ZZY = Vo 9;! E4, 02(J) = FFY = FX = C1(Vo), [E, J] ::; Vo,
and for some r, 02( J) / E is the direct sum of r natural modules for J / 02 ( J) with
[02(J)/E, U] = F/E. Thus
J has r + 1 noncentral 2-chief factors.
Moreover J and E are normal in Na(Vo).
As 02 (J)/E is abelianand 02 (J) = XF, [X, F]::; E. Similarly as [XF/E, U] =
F/E and JU : FJ = 2, for u E U - F the map cp : X/E ___, F/E defined by
cp(xE) := [u, x]E is a bijection. Therefore as [U, QH] = Z ::; E by 15.3.52 and
15.3.48.2, X n QH = E. Finally cI>(E) ::::;; cI>(U) n cI>(UY) = z n ZY = 1, so by (1), E
is totally singular in the orthogonal space U.
Next J = Y1U 9;! 83 x Z2, with F = X = Z(J) = Un N 2 ; in particular
[z.L,X] = Z. By 15.3.54.4, Y1 has s noncentral chief factors on P/V, and by
15.3.53.1, Y 1 has two noncentral chief factors on V. Thus J has s + 2 noncentral