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.