1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

2.4. THE CASE WHERE rQ IS NONEMPTY 527 Thus 02(H n M):::; 8 and CG(0 2 (H)):::; H, so Co2(M)(8):::; Co2(M)(02(H)):::; 02(M) n H: ...

528 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 2.4.2, S1 E Syl2(H1) and S1 EU. Then as IH1l2 ::'.": IHl2 = ISi by hypothesis, we ...

2.4. THE CASE WHERE rg IS NONEMPTY 529 2.4.1. Shadows of groups of rank 2 with L 2 (2n)-blocks. In this subsec- tion we continue ...

530 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 by B.4.2.1, and hence R E Syl 2 (LQ). Then (3) will follow once we prove (2). Howe ...

2.4. THE CASE WHERE rQ IS NONEMPTY 531 determined up to isomorphism in each case. The parabolics P in cases (a) and (b) of (4) e ...

532 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 £X = Nx(Qx) since Nx(Qx) ~ Nx(Q) = L. Thus Z(L) = P = Z(LX), contrary to 2.4.4. Th ...

2.4. THE CASE WHERE r~ IS NONEMPTY 533 (1) Z(L) = 1, Z(R) = CR(t), R is transitive on t[R, t], and [R, t] is transitive on tZ(R) ...

534 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 If Z(L) = 1 then z = zx by (2); hence (zx)L = zL gives the partition of the natura ...

2.4. THE CASE WHERE r~ IS NONEMPTY LEMMA 2.4.20. Assume Z(L) =f 1. Then for x ET-S: (1) B = D x DX is regular on 6. := Z(R) - (Z ...

536 2. CLASSIFYING THE GROUPS WITH IM(r)I =^1 so Z(I) = [I,I] ::; R*. Now we saw that involutions of Z(I) lie in~' so we may ass ...

2.4. THE CASE WHERE I'Q IS NONEMPTY 537 At this point, we have obtained strong control over the 2-local structure and 2-fusion o ...

538 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 possibilities in Theorem C (A.2.3) that Ks = K ~ Sp4(q^112 ), and x induces an out ...

2.4. THE CASE WHERE r3 IS NONEMPTY 539 to show that s^0 n T ~ s, for the involution we have been denoting bys: For then s^0 n Tc ...

540 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 We can now argue much as in the proof of 2.4.8.5, but using M 0 , Lo in the roles ...

2.4. THE CASE WHERE q IS NONEMPTY S41 {R1, Rz} transitively, so that INT(R1)I 2:: 2181 = 21821 > 182 1. This contradiction el ...

542 2. CLASSIFYING THE GROUPS WITH IM(T)I = i By C.5.5, we may choose x E Mo with U"' 1:. Q, and and so as Qi = Q2 with Q = Qin ...

2.4. THE CASE WHERE r 0 IS NONEMPTY 543 S1 E Syb(X) with T1 ::; S1, we have L E .C(X, S 1 ). This is a contradiction, since from ...

544 2. CLASSIFYING THE GROUPS WITH IM(T)I = i it remains to show that A(S) = A(T), or equivalently to establish the assertion J( ...

2.4. THE CASE WHERE r~ IS NONEMPTY 545 Set Gz := Cc(Z) and Gz := Gz/Z. Then Pi'c = J(S) is x-invariant, so PTc :::;! (x,KS) =: M ...

546 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 of order 27 and rank 4, and Hu =Na,,. (U) ~ Ds x S4, to conclude that K ~ A5, L 2 ...