1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

1.2. THE SET .C(G, T) OF NONSOLVABLE UNIQUENESS SUBGROUPS 507 Assume as in (2) that L -=J L8, and that L < K; then Ks -=J K b ...

508 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS LEMMA 1.2.11. Let H E 1i with T n H =: TH E Syl2(H), and K E C(H). Assu ...

1.3. THE SET B*(G, T) OF SOLVABLE UNIQUENESS SUBGROUPS OF G 509 and 02(X) = [02(X), P], giving conclusion (2). Notice XT =PT, so ...

510 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS R = P-or R ~ Zp, P ~ p1+^2 , and R = Z(P). In particular by A.1.21, P c ...

1.3. THE SET 8* (G, T) OF SOLVABLE UNIQUENESS SUBGROUPS OF G 511 "K" in that result, so that P n C1(Z(f'))^00 is cyclic oforder ...

512 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS We have the following corollaries to Proposition 1.3.4: PROPOSITION 1.3 ...

1.3. THE SET 8*(G, T) OF SOLVABLE UNIQUENESS SUBGROUPS OF G 513 Next we obtain some restrictions on L. If R :':! LT then any 1 = ...

S14 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS Furth~rmore HE He by 1.1.4.6. So as K f:. Mand R acts on K, we have the ...

1.4. PROPERTIES OF SOME UNIQUENESS SUBGROUPS {2) Q E l!IG(Lo, 2) = Syb(CM(Lo/02(Lo))). {3) Ca(Q) ~ 02(M) ~ Q. (4) If LE Xf, then ...

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CHAPTER 2 Classifying the groups with IM (T) I = 1 Recall from the outline in the Introduction to Volume II that the bulk of the ...

518 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 2.1. Statement of main result Our main theorem in this chapter is: THEOREM 2.1.1. ...

2.2. BENDER GROUPS 519 PROPOSITION 2.2.2. Assume for each D in the Alperin-Goldschmidt conjuga- tion family that Na(D) ::; M for ...

520 2. CLASSIFYING THE GROUPS WITH [M(T)[ = 1 groups in 1.1.5.3, Lis a Bender group with Autu(L) = S11(S n L). In particular, U ...

2.3. PRELIMINARY ANALYSIS OF THE SET ro 521 PROOF. Let 1) be the Alperin-Goldschmidt conjugation family for Tin G. By 2.2.3, D ~ ...

522 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 CH(Q):::; Q, so Q E Si,(G) by 1.1.4.3. In particular applying this observation to ...

2.3. PRELIMINARY ANALYSIS OF THE SET ro 523 PROOF. Let U E U be of maximal order and Hu E 'He(U, M). Then JVI :S:: IUI ::::: IHl ...

524 s. CLASSIFYING THE GROUPS WITH IM(T)I = 1 (b) If J(S) s Rs S with IS: RI= 2 and Co 2 (M)(R) SR, then RE (3. (c) If H E 1-i ...

2.3. PRELIMINARY ANALYSIS OF THE SET ro 525 It remains to prove (6). By (1), S :::; M and S E Syb(H n M). Assume that H E I'o. T ...

526 2. CLASSIFYING THE GROUPS WITH IM(T)I = i Similarly if L is a component of H, then by 1.1.5.3, L = [L, z] 1:. M, and the pos ...