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.
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
( 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.
1.1. G is of even type if and only if
1. £2(G) c e
2. 02'(CG(X)) = 1 for every involution x G G; and
3. G has 2-rank at least 3.
1.2. G is of small odd type if and only if G is not of even type
andH2{G) n S
= 0.
Thus, if G is of small odd type, and rri2(G) 3, then one of the following holds:
(1) JC
(G) fl T2 ^ 0 = £
(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
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