16.5. IDENTIFYING Ji, AND OBTAINING THE FINAL CONTRADICTION 1197conclusion (iv) of that result holds: namely, m(V) = 3 and AutM(V) ~ Frob 21.
But then M acts irreducibly on V, impossible as M acts on cI>(J(T)) = VL. D
NOTATION 16.5.5. For u E U#, define Xu := 03 (Lu) if L ~ L 3 (2n), n even,
and Xu := Lu otherwise. In the former case, n > 2 by 16.5.4. Thus in any event,
Xu S L^9 S Ca(R) by 16.4.8, so R :::; CH (X~). Further for i an involution in
Tc, we can define xi :::; LB analogously. Then Xu :::; xi as Xu ::; CLY(i), so by
symmetry between L, u and L^9 , i, Xi s Xu. Thus we may define X := Xu = Xi.
Inspecting the possibilities for L* remaining in (E2) after 16.5.1, 16.5.3, and16.5.4, we conclude from 16.1.4 and 16.1.5 that for each involution j in L, Xj i=- 1
except when L ~ Sp4(2n) with n > 1, and j* is of type c 2 •
Observe that the fourth part of the next lemma supplies another assertion
about the symmetry between Kand K'.
LEMMA 16.5.6. Let fr' := H' / K'.
(1) X =Xu= Xi for all involutions i E Tc and u ER.
(2) R* s CL*r•(X*).(3) If we choose choose g as in 16.4.2.4, then g E Na(X).
(4) The following are equivalent:
{a) Some involution in R* is 2-central in L*.
{b) Each involution in Tc is 2-central in L'.{c) Some involution in Tc is 2-central in L'.
( d) Each involution in R* is 2-central in L *.(5) Assume that Z(L) = 1, and for each J E .6.. 0 and each involution i in
NJ ( K), that i is not 2-central in L . Let v be the projection of u on L,. and
suppose there is l E L with vv^1 an involution of X. Then vv^1 is not 2-centra,l in L'.
PROOF. We already observed that (1) and (2) hold. We saw in 16.4.11.3 that
if we choose g as in 16.4.2.4, then Tfj = R, so (1) implies (3).
Suppose u is 2-central in L. By (1), for each involution i E Tc, Xu= Xi,
so by inspection of the centralizers of involutions of H* listed in 16.1.4 and 16.1.5
remaining after 16.5.1, 16.5.3, and 16.5.4, we conclude that I is also 2-central in
L'. Thus (4a) implies (4b). Then as K E .6..o(K') by 16.4.11.2, by symmetry (4c)
implies (4d). Of course (4b) implies (4c), and (4d) implies (4a), so (4) holds.
Assume the hypotheses of (5). As Z(L) = 1, u = jv for some j E Tc with
j^2 = 1, so uu^1 = (jv)(jv^1 ) = vv^1 EX:::; L' by hypothesis and (1), and v/ = iHi. Let
i := u^1 B-
1and J := (K')^1 B-
1. As uu^1 EX while g E Na(X) by (3), uB-
1
u^1 B-
1
EX;
thus as uB-
1
EK, i = u^1 B-
1
E Jn XK, so J E .6.. 0 by 16.4.9.3. By hypothesis i is
not 2-central in L , so conjugating by g, vv^1 = v/ is not 2-central in L', establishing
(5). D
LEMMA 16.5.7. Assume Z(L) = 1, !Cr• (TL)I = 2, and IT*: T£1S2. Then
{1) R contains no 2-central involution of L.
(2) L has more than one class of involutions.
PROOF. As U:::; LK while Z(L) = 1 by hypothesis, (1) implies (2). Hence we
may assume that (1) fails, and it remains to derive a contradiction.
By 16.5.6.4, the hypotheses of 16.5.2.2 are satisfied. If case (a) of 16.5.2.2 holds,