1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

16.4. INTEB.SECTIONS OF NG(L) WITH CONJUGATES OF CG(L) 1187 Since we saw 02(Lr) ~ L', we conclude by symmetry that 02 (Lr) :::1 ...

1188 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC 03 ,(Lr) = 1, Lr is faithful on L' and Lr ::l 02 (Cb,(z)). We ...

16.4. INTERSECTIONS OF Na(L) WITH CONJUGATES OF Ca(L) 1189 We will first show in 16.4.9.2 that Ao is nonempty. The shadow of 810 ...

1190 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC (+) If r^1 = rv for some l EL and 1 i= v E .CL(r), then r^1 a ...

16.4. INTERSECTIONS OF NG(L) WITH CONJUGATES OF CG(L) 1191 [To[ > 2, we showed z EL and L* is not L3(4) or M2~. So in any cas ...

1192 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC Therefore 02 (L) = 1, so L ~ L 3 (4). Then Q centralizes E, a ...

16.4. INTERSECTIONS OF NG(L) WITH CONJUGATES OF CG(L) 1193 particular Tc does not centralize R. For 1 "I= r E R, observe by 16.4 ...

1194 i6. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC Choose g as in 16.4.2.4; that is, so that Nr(K') :::; TB. The ...

16.5. IDENTIFYING J1, AND OBTAINING THE FINAL CONTRADICTION 1195 Thus L ~ £ 2 (4). Then H = LK by 16.4.6, so R induces inner aut ...

1196 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC U = !1 1 (R) is of order 2, so as R ~Tc, m2(R) = 1 = m2(Tc). ...

16.5. IDENTIFYING Ji, AND OBTAINING THE FINAL CONTRADICTION 1197 conclusion (iv) of that result holds: namely, m(V) = 3 and AutM ...

1198 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC (b) of 16.5.2.2 holds, and replacing T by the subgroup T^1 de ...

16.5. IDENTIFYING Ji, AND OBTAINING THE FINAL CONTRADICTION 1199 (6) Assume that Z(L) = 1, v is the projection of u on L, and th ...

1200 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC Therefore L* is M 12 , J 2 , HS, or Ru. If Z(L) -=/:-1, then ...

16.5. IDENTIFYING Ji, AND OBTAINING THE FINAL CONTRADICTION 1201 L', Tc and L, R, CM(u) moves z, so zM is of order 7 or 13. Furt ...

1202 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC is irreducible on V and M normalizes X containing Z, V:::; X: ...

16.5. IDENTIFYING J1, AND OBTAINING THE FINAL CONTRADICTION 1203 (b) L ~^2 F4(2n)', X/02(X) ~ L2(2n)', Z(02(X)) = V EB W, where ...

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