2. THE SPECIAL TYPE HYPOTHESIS
3
(5) If p = 5, K is not isomorphic to
2
F
4
(2i)' ,
2
F
4
(2§), J
2
, Coi, Co2,
Co3, HS, Mc, Ly, Ru, He, Fi22, F
5
, F3, F
2
, or F
i ;
(1C) (6) If p = 7, K is not isomorphic to Cox, He, ON, Fi'2A, F3 or Fi;
and
(7) I f p = l l , K^J4.
A nonsimple !X-group K with Op(K) = 1 is called a Sp-group provided one of
the following holds:
(1) K/Z(K) e %, but K ^ 2An, n = 9, 10, or 11, with p = 2;
(ID) (2) K 9* Sp4{3), SL4{3), or SU4{3) and p = 2; or
(3) K ^ 307V and p = 3.
[The groups PSp
4
(3), L
4
(3) and £/"4(3) are not 92-groups, but are C2-groups.]
In particular, for p 2, the groups in Alt which are 32-groups are the groups
An, n 9, and 2An, n 12. Likewise, the groups in Spor which are S2-groups are
J i , Mc, Ly, ON, Co3, He, and F6.
An equivalent list of the Sp-groups, but sorted according to the union Qhev U
Alt U Spor rather than by the prime p, is given in Chapter 6 (I A).
Finally, recall from [I2; 3.1] that the set of target groups for Theorem 67 is
the set %(7\ consisting of the alternating groups An, n 13, together with all
groups in Qhev with some low rank exceptions [I2; pp. 86-87]. The weakening of
the hypothesis p G cr0(G) when p is odd to the condition (IB) entails enlarging
this target set to include the groups U$(2n), n 1, which were formerly excluded
because mv(U^(2n)) 4 for all odd primes p. Thus slightly expanded, our target
set is the following set of simple groups (here e takes the values ± 1 and q is a prime
power)
K(7)* = {An I n 13}
U {Aen(q) I n 5} U {A\(q) \ e = 1 or q odd} U {Ae3(q) \q = e mod 8}
U{Bn(q)\n 3, q odd}
U {Cn(q) \n 4} U{Cs(q)\q odd}
U
{Den(q)
I n 5} U {D\{q) \ e = 1 or q odd}
U{F
4
(g)}
U{E^(q),E7(q),E8(q)}
- {A4{2), A%{2), A%(2), C
4
(2), D
4
(2), D5"(2), F
4
(2), B6"(2), B
3
(3), /?|(3)}.
2. T h e Special T y p e Hypothesi s
Before stating our theorem we need to formalize the statements of the special
type results to be added to its hypothesis. These results concern elements x G Tp(G)
such that CG(X) contains a p-component K with K being p-terminal in G [1^; 6.26]
and K/Op'(K) a Tp-group [I2; 13.1]. They involve slight extensions of Theorems C2
and 66, which will be established in the course of the analysis of Parts IV and V.
For brevity, if x and K are as just specified, we call (x, K) a p - t e r m i n a l T
p
-pair.
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