560 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1Observe that by 2.5.7 and 2.5.15, we have reduced the list of possibilities for
K in 2.5.2 to:
LEMMA 2.5.16. One of the following holds:
(1) K ~ L 2 (p), p > 7 a Mersenne or Fermat prime.
(2) K ~ L3(2) and NH(K)/Cs(K) ~ Aut(L3(2)).
(3) K ~ A5 and NH(K)/Cs(K) ~ M10, PGL2(9), or Aut(A5).
REMARK 2.5.17. All of these configurations appear in some shadow which is of
even characteristic, and in which a Sylow 2-group is in a unique maximal 2-local.Usually the shadow is even quasithin. The group is not simple, but it takes some
effort to demonstrate this and hence produce a contradiction.
The groups L2 (p) x L2 (p) extended by a 2-group interchanging the components
are shadows realizing the configurations in (1) and (2), while L 4 (3) ~ POci(2)
extended by a suitable group of outer automorphisms realize the configurations in
(3). The last case causes the most difficulties, "and consequently is not eliminated
until the final reduction.LEMMA 2.5.18. (1) K is a component of Gt.
(2) Gt= KoSCa.(Ko) with Ca.(Ko)S::::; M.
(3) Ca.(Ko) E 7-le, so O(Gt) = 1.PROOF. By 2.5.11.3, H :S Gt E I'* and S E Syb(Gt). Thus if K is not a
component of Gt, we may apply 2.5.6 with (t) in the role of "E", to conclude that
K =Ko~ A5 and Kt := (K^0 •) ~Mn. Since HE r+ n I'*, we conclude from
parts (2) and ( 4) of 2.3. 7 that KtS E r+ n I'*, contrary to 2.5.15.
Thus (1) holds, so as S E Syb( Gt), Ko::::! Gt, and by 2.5.8.1, Gt= KoSCa. (K 0 ).
Then Ca.(Ko) :S Ca.(zK) :S Ca.(z)::::; M, proving (2). By 2.3.9.4, Gt n ME 7-le,
so (2) implies (3). DLEMMA 2.5.19. Assume i is an involution in Cs(K) such that K is not a
component of Ca(i). Then
(1) K =Ko.
(2) Cs(i) n Cs(K) = (t, i).
(3) There exists a component Ki of Ca(i) such that either:
(I) Ki =J Kf, K = CKim(t)'~", and Ki~ K ~ L2(P), p 2 7, or
(II) K = CKi (t)^00 , and ~ne of the following holds:
(a) K ~ L3 (2), and t induces a field automorphism on Ki ~ L 3 ( 4) or
(b) K ~ L3(2), and t induces an outer automorphism on Ki~ J2.
(c) K ~ A6 and Ki~ Sp4(4), L5(2), HS, or As.
(4) Either z = ZK EK and tz E t^0 , or Ki~ As and t induces a transposition
on Ki.
PROOF. Let Gi := Ca(i) and R :=Gin C 8 (K). Ast E Z(S) n Sc, (t, i) :SRby our hypothesis on i. As K is not a component of Gi, i =J t by 2.5.18. Therefore
i tf. Z(S), or otherwise i centralizes (K^8 ) = K 0 , whereas Z(S) n Sc = (t) by
2.5.11.1. By 2.5.18, Ca.(Ko) ::::; Mand SE Syb(Gt), so conjugating in Ca.(K 0 )