CHAPTER 1

General Introduction to the Special Odd Case

1. The Goals: Theorems C2 and C3

As always in this series,

G is a X-proper simple group.

This volume is devoted primarily to the proof of the following theorem.

THEOREM

C2 U C3. Let G be of special odd type. Then G is isomorphic to one

of the following simple groups:

(a) L2(q), q odd, q 5;

(b) L%(q) or PSp4{q) or G2(q), q odd, q 3, e = ±1;

(c) *D±(q), q odd;

(d) L\(q), q odd, q 3, and q ^ e (mod 8);

(e) 2 G

2

( 3 2 n + 1 ) , n l ;

(f) A7, A9, A10 or An;

(g) Afn or J73(4); or

(h) Mi2f Ji, Mc, Ly, orO'N.

This theorem combines Theorems 62 and 63, as stated in Sections 20 and 21

of [IG]. For the sporadic groups in (h), we prove that G « G*, which means that G

and G* have the same centralizer of involution pattern. This means that there is

an isomorphism x — * x* from a Sylow 2-subgroup S of G onto a Sylow 2-subgroup

5* of G* preserving conjugacy of involutions and isomorphism type of involution

centralizers (cf. [I2; Section 19]). By an assumed Background Result [Ii; (16.1)],

this implies that G = G*.

In the statement we use the notational convention that e = ±1 or (or simply

±), and L+(q) = Ln(g), while L~(q) = Un(q). In order to explain the special odd

type hypothesis, we need some definitions.

DEFINITION

1.1. G is of even type if and only if

1. £2(G) c e

2

;

2. 02'(CG(X)) = 1 for every involution x G G; and

3. G has 2-rank at least 3.

DEFINITION

1.2. G is of small odd type if and only if G is not of even type

andH2{G) n S

2

= 0.

Thus, if G is of small odd type, and rri2(G) 3, then one of the following holds:

(1) JC

2

(G) fl T2 ^ 0 = £

2

(G) H g2; or

(2) C2(G) C S2 and

0 2 ' ( G G ( # ) )

7^ 1 f°r some involution x G G.

A corollary - and simplified version - of our main theorem is the following

theorem.

1

http://dx.doi.org/10.1090/surv/040.6/01