CHAPTE R 1

Theorem CV: General Introduction

1. Generic Simple Group s

For expository reasons, we have chosen to classify the generic simple groups

before treating the special ones, even though the logical development of the proof

as described in the classification grid [I2; p. 85] proceeds in the reverse order. Thus,

in effect, we establish Theorem C7 on the assumption that Theorems C^, 1 i 6,

have been proved.

The specific results from these special type theorems needed for the proof of

Theorem 67 will be added to its hypothesis, and will be specified in the next section.

We then formulate a precise Theorem 67 including this feature.

In any event, Theorem C7 classifies the 3C-proper simple groups G of generic

type. By the definition of "generic type" [I2; 16.1], either (a) some element of ^(G)

is a S2-group, or (b) G is of restricted even type, cr0(G) is a certain nonempty set

of odd primes, and for any p G o"o(G), some element of £°(G ) is a Sp-group. Given

G, we begin by fixing a prime p as follows: in case (a) we take p = 2, while in case

(b) we take any p G cro(G).

Indeed in Part III we shall use only certain properties of the primes in the set

7o(G) in case (b), and the theorem will be stated and proved so that in case (b)

it applies to all primes with those properties. For example, the extra requirements

for a group of even type to be of restricted even type [I2; 8.8] are of no consequence

in this part. Hence we shall assume that

(1A) If p is odd, then G has even type.

By definition [I2; 14.1], £p(G) is the set of all components of all the groups

CG{X)/OP(CG(X)), where x ranges over the elements of Z£(G)—that is, x is any

element of G of order p for which 7TIP{CG{X)) 3 or 4 according as p — 2 or p is

odd.

Furthermore by definition [I2; 1.11], (JQ(G) is defined when G is of restricted

even type and consists of those odd primes for which an appropriate 2-local sub-

group NofG has p-rank at least 4, and for which G possesses no strong p-uniqueness

subgroup. For Part III, it is only the latter feature which matters, and we shall dis-

regard the subgroup N and replace the condition p G cro(G) by a weaker condition.

Indeed there are several types of strong p-uniqueness subgroup, according to the

definition [I2; 8.7], and we only need to assume that G has no strong p-uniqueness

subgroups of certain types.

D E F I N I T I ON 1.1. A subgroup M of G is a strong p-uniqueness subgroup of

component type if and only if

(a) M is ap-component preuniqueness subgroup of G, and mp(M) 4 if p 2;

(b) M is almost strongly p-embedded in G; and

http://dx.doi.org/10.1090/surv/040.5/01