This page contains documentation relating to the
classification of simple K*-groups of finite Morley rank
of odd and degenerate types.
Theorem
Let G be a simple K*-group of finite Morley rank of odd type. Then one of the following holds:
- G is algebraic;
- G has normal 2-rank at most 2;
- G is a minimal connected simple group (i.e., G has no proper definable connected simple sections; that is, G is minimal simple, apart from the possible involvement of some finite simple sections).
One begins with the following "generic" identification theorem, which is the case p=2 of a result given in
Theorem Let G be a simple K*-group of finite Morley rank and of odd type. Suppose that G has Prufer 2-rank at least 3, and let D be a maximal 2-torus. Assume that
Then G is a Chevalley group over an algebraically closed field.
- G is generated by the connected components of centralizers of involutions of D;
- For each involution i in D, the connected component H of its centralizer is reductive: that is, of the form H=Z(H)E(H).
This can be put into a more convenient form using the notion of the 2-generated core of G, relative to a 2-Sylow subgroup S. This is the definable closure of the group generated by normalizers of elementary abelian subgroups of S of rank 2. Taking into account some intermediate results of
The last of these cases can be eliminated under the assumption of Prüfer rank at least 3, or even assuming normal rank at least 3. This is done using the signalizer functor method. This is done in the tame case in Borovik's article cited above, using the fact that in that case O(C(i)) must be nilpotent. The general case is handled by a recent result of Jeff Burdges (article in preparation as of August 2002) showing that any nontrivial solvable signalizer functor gives rise to a nilpotent one.
We have also to deal with case (3).
Theorem Let G be a simple K*-group of finite Morley rank, of odd type, and of Prüfer 2-rank at least 3. If the 2-generated core relative to a Sylow 2-subgroup of G is a proper subgroup, then G is a minimal group. This is proved in
- A. Borovik and A. Nesin
- 2-generated cores in groups of finite Morley rank of odd type, preprint (Summer 2002).
Refining the bound on Prüfer rank to a bound on normal rank requires a further argument which is found in A new trichotomy theorem for groups of finite Morley rank of odd or degenerate type by A. Borovik.
None of the above uses tameness. However in the tame case, it is known that minimal groups of odd type have Prüfer 2-rank at most 2:
If G is degenerate, this can be simplified: G is either algebraic, or has proper 2-generated core, or has Prufer 2-rank at most 2.
Theorem
Let G be a simple K*-group of finite Morley rank of degenerate type. Then one of the following holds:
- G is algebraic;
- G has proper 2-generated core
- G has normal 2-rank at most 2;
This is parallel to the odd type case, but much simpler. In the degenerate case, a simple K*-group is minimal, and in particular if H is the connected component of an involution then O(H)=H>1. So we need only the portion of the argument corresponding to the use of signalizer functors in Borovik's article, extended by Burdges' result. (This possibility was not actually mentioned in Borovik's article, since in the tame case the question of degenerate type groups does not arise.)