2 1. GENERAL INTRODUCTION TO THE SPECIAL ODD CASE

THEOREM

62,3- Let G be a group of small odd type. Then G is as described in

the conclusion of Theorem 62 U C3, but G is not isomorphic to Ag, A\Q, M12, or

We remark that Ag, Aio, M\2 and J\ all satisfy the Aschbacher-Smith condi-

tions to be quasithin groups of even type and, as such, are identified in their paper

[ASml].

The condition of special odd type is a slight broadening of the condition of small

odd type to permit certain groups of even type in which an involution centralizer

has a component K with K = L2(q) for some q G FM9 or with K/Z{K) = L3(4).

Specifically we need the following technical definition.

DEFINITION

1.3. We say that G is of restricted even type if and only if G is

of even type and the following conditions hold:

(a) IfKe H2(G) with K^L2(q), q odd, then q G {5,7,9,17} and m2(G) 4;

(b) Suppose that x is an involution of G and K is a terminal component of

CG{X) such that K = L2(q), q odd. Then the following conditions hold:

(1) m2(CG(x)) 4;

(2) x is not 2-central in G; and

(3) Either q = 5 and NQ(K) has Sylow 2-subgroups isomorphic to E24, or

q = 9 and m2(CG(K)) = 1; and

(c) Suppose that x and y are commuting involutions of G, K is a terminal

component of

CG(%),

K/Z{K) = Ls(4), and y induces an automorphism, of

unitary type on K. Then \G :

NG{K)\2

4.

Now we can define special odd type.

DEFINITION

1.4. G is of special odd type if G is not of restricted even type and

£2(G)ng2 = 0.

Last, we remark that we in fact prove a somewhat stronger theorem than The-

orem C2, which we call Theorem Q2 and which is needed for the proof of Theorem

67. We shall state this theorem in the next section.

2. Theorems C2 and C£

We now separate our discussion of Theorem 62 from that of Theorem 63. In

this section we discuss Theorems 62 and C2. First we state Theorem C2.

THEOREM

62- Suppose that G is of £j*B2-type and contains no 2-uniqueness

subgroup. Then G satisfies one of the conclusions of Theorem Q2 U C3 and G ^

U3(4), Jx, orAnforne {5,9,10,11}.

The exclusion of A*,, U3 (4) and J\ from the conclusion of Theorem C2 is due to

the hypothesis that G contains no 2-uniqueness subgroup. The other alternating

groups will arise as conclusions to Theorem 63. As in [I2], we denote by /C^ the

set of simple groups arising as conclusions to Theorem Q2.

We now explain the terminology of Theorem Q2 and comment on reformulations

and generalizations.

The meaning of £!B2-type. Our entire analysis rests on the subdivision of

the set %2 of all known quasisimple groups K with Z(K) = 02{K) into three

subsets. We remind the reader of the relevant definitions beginning with the sets

62, T2, and S2 from [I2; Sections 12-14], along with the subset (B2 ofT2.