1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

Part 7 The Even Type Theorem ...

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CHAPTER 16 Quasithin groups of even type but not even characteristic The original proof of the Classification of the finite simp ...

1170 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC The definition of even type is given on p.55 of [GLS94]. We w ...

16.1. EVEN TYPE GROUPS, AND COMPONENTS IN CENTRALI:ZERS 1171 NOTATION 16.1.3. Recall that the types of twisted groups in Theorem ...

1172 i6. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC {8} L ~ J 2 and either r is inner with CL(r) ~ As/Qs * Ds or ...

16.2. NORMALITY AND OTHER PROPERTIES OF COMPONENTS 1173 PROOF. Let R := K/02(K). Since Lis a component of Ca(t) and i centralize ...

1174 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC PROOF. Let CT(t) ::; S E Syl2(CG(t)), so that CT(t) ::; Cs(z) ...

16.2. NORMALITY AND OTHER PROPERTIES OF COMPONENTS 1175 PROOF. Let g E Na(Lu), and let t be an involution in Lu. By 16.2.6.1, Li ...

1176 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC Then since Ho ::; H < G and TL 1:. T£, conclusion (3) of 3 ...

16.3. SHOWING L IS STANDARD IN G 1177 In case (ii), CL 0 (i) ~ Dp+E x Dp+E, where p = E mod 4 and E = ±1. But 3 divides p + E as ...

1178 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC Observe also (cf. I.7.2.5): REMARK 16.3.3. If Lis standard in ...

16.3. SHOWING L IS STANDARD IN G 1179 as E is of order 4, so by hypothesis K is a component of the centralizer of an involution ...

1180 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC L is not a component of Gt. Therefore Cs(t) = (z), so as Sis ...

16.3. SHOWING L IS STANDARD IN G 1181 by I.2.2.7b, contrary to(!!). Thus K ~ G 2 (4). By I.2.2.5a, 2-central involutions of K li ...

1182 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC while m 2 (Tc) = 2 by 16.3.8; hence m2(T):::; m2(T/Tc) + m2(T ...

16.4. INTERSECTIONS OF Na(L) WITH CONJUGATES OF Ca(L) 1183 In this section we develop some technical tools, which we apply in th ...

1184 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC LEMMA 16.4.3. {1) KE b..(K'). {2) L' = [L',z]. PROOF. Part (I ...

16.4. INTERSECTIONS OF Na(L) WITH CONJUGATES OF Ca(L) 1185 We claim that z is weakly closed in Z(T) with respect to G; the proof ...

1186 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC is transitive on V - (vz), which is impossible since v tfi z^ ...