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