4 PART III, CHAPTER 1. THEOREM C7: GENERAL INTRODUCTION

(2A)

We recall that by definition [I2; 13.1], a group L is a Tp-group if and only if

L G %p and one of the following holds:

(1) p = 2 and L ^ SL2(q), q odd, q 5; L2(q), q odd, q £ JM9;

(X)L

3

(4), X = 4, 4 x 2, or 4 x 4; A7; or 2A

n

, 7 ra 11;

(2) p is odd and mp(L) = 1 with L not a Cp-group (in particular,

L ^ L

2

( 8 ) Up = 3 a n d L ^ 2 £

2

( 2 § ) ifp = 5);

(3) p = 3 and L/Z(L ) ^ L|(g), q = e mod 3, e - ± 1 ; A7; M

i 2

, M

2 2

,

or J

2

; or

(4) p = 3 and L ^ 3,46; or

(5) p = 5 and L ^ Fi22.

The nonsimple Tp-groups are called 'B^-groups. Thus L is a 3

p

-group if and

only if one of the following holds:

(1) p = 2 and L ^ SL

2

(g), Q odd, g 5, (X)L

3

(4), I = 4 , 4 x 2 , o r

(2B) 4 x 4, or 2An, 7 n 11; or

(2) p = 3 and L ^ SL|(g), 4 = e mod 3, e = ± 1 , 3A6, 3A7, or 3M

2 2

.

Also if p is odd, we shall call a Sp-group L a TS

p

-group provided either

mp(L/Z(L)) 4, or p = 3 and L ^ E/7(2). The Tg

p

-groups are "close" to 7p-

groups in the sense that most of them are pumpups of T^-groups.

Now we are in a position to state the extensions of Theorems C2 and QQ needed

for the proof of Theorem 67. These results assert under appropriate hypotheses

that either the given group G is known—isomorphic to a group in %2 or XQ (see

[I2; 3.1])—or else G contains a strong p-uniqueness subgroup. The hypotheses are

captured in the following definition.

D E F I N I T I ON 2.1 . Let G be a ^-proper simple group and p a prime. We say

that G has a p-Thin Configuration if and only if

(a) G contains a p-terminal Tp-pair (x,K);

(b) If p = 2, then Kj02\K) lies in £

2

; and

(c) If p is odd, then

(1) Every element of ££(G) which is in S

p

p lies in TS

p

;

(2) If some element of £JZ(G) lies in T9

P

and has p-rank at least 3, then

p — 3 and mp(K) 2; and

(3) If p = 3, J e H°3(G) is a Tg

3

-group, y G T

3

(Aut(J)) and Ly(Cj(y)) has

a 3-component / such that 1/0^(1) is a ^ - g r o u p , then K/Oy{K) is

itself a 23-group.

We may also say that (x,K) affords a p-Thin Configuration for G.

The results we assume from Theorems C2 and QQ assert that if G has a p-Thin

Configuration, then either G G % or G possesses a strong p-uniqueness subgroup.

Since we are already assuming that G has no strong p-uniqueness subgroup, we

may and shall proceed under the assumption that

(2C) G has no p-Thin Configuration.

3. Statemen t of T h e o r e m C£

Now we can state in precise form the principal result to be proved in Part III.