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
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.
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