1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

482 INTRODUCTION TO VOLUME II

where M(T) denotes those members of M containing T. The class of simple groups

of even characteristic contains some families in addition to the groups of Lie type

in characteristic 2. In particular it is larger than the class of simple groups of

characteristic 2-type (discussed in the next section), which played the analogous

role in the original proof of the Classification.

The Classification proceeds by induction on the group order. Thus if G is a

minimal counterexample to the Classification, then each proper subgroup H of G

is a K-group; that is, all composition factors of each subgroup of H lie in the set /(,

of known finite simple groups.

Finally quasithin groups are "small" by a measure of size introduced by Thomp-

son in the N-group paper [Tho68]. Define

e(G) := max{mp(M): ME Mand pis an odd prime}

where mp(M) is the p-rank of M (namely the maximum rank of an elementary

abelian p-subgroup of M). When G is of Lie type in characteristic 2, e(G) is a

good abstract approximation of the Lie rank of G. Janko called the groups with

e( G) ~ 1 "thin groups", leading Gorenstein to define G to be quasithin if e( G) ~ 2.

The groups of Lie type of characteristic 2 and Lie rank 1 or 2 are the "generic"

simple quasithin groups of even characteristic.

Define a finite group H to be strongly quasithin if mp(H) ~ 2 for all odd primes

p. Thus the 2-locals of quasithin groups are strongly quasithin.

We combine the three principal conditions into a single hypothesis:

Main Hypothesis. Define G to be a QTKE-group if

(QT) G is quasithin,

(K) all proper subgroups of G are K-groups, and

(E) G is of even characteristic.

We prove:

THEOREM 0.1.1 (Main Theorem). The finite simple QTKE-groups consist of:

(1) (Generic case) Groups of Lie type of characteristic 2 and Lie rank at most

2, but U5(q) only for q = 4.

(2) (Certain groups of rank 3 or 4) L4(2), L5(2), Sp5(2).

(3) (Alternating groups:) A5, A5, A 8 , Ag.

(4) (Lie type of odd characteristic) L 2 (p), p a Mersenne or Fermat prime;

L3(3), LH3), G2(3).

(5) (sporadic) Mu, M12, M22, M23, M24, J2, J3, J4, HS, He, Ru.

We recall that there is an "original" or "first generation" proof of the Classifi-

cation, made up by and large of work done before 1980; and a "second generation"

program in progress, whose aim is to produce a somewhat different and simpler

proof of the Classification. The two programs take the same general approach, but

often differ in detail. Our work is a part of both efforts.

In particular Gorenstein, Lyons, and Solomon (GLS) are in the midst of a

major program to revise and simplify the proof of part of the Classification. We

also prove a corollary to our Main Theorem, which supplies a bridge between that

result and the GLS program. We now discuss that corollary:

There is yet another way to approach the characterization of the groups of Lie

type of characteristic 2. The GLS program requires a classification of quasithin