The Even Type Theorem

This page contains documentation relating to the Even Type Conjecture (now, Theorem)

Update

The material below is superseded by a book which proves the full result without the "K*" hypothesis (as first envisioned in Altinel's Habilitation). The book is

• Simple Groups of Finite Morley Rank of Even Type
with T. Altinel and A. Borovik, Mathematical Surveys and Monographs, vol. 145,
American Mathematical Society, Providence, RI, 2008
xx+556 pp.

File:

"Final paper"

This paper contains a statement of the conjecture and a proof in the so-called K*-case modulo various ingredients. This was written well before the papers on which it relies, and has been revised as the situation evolved, though not proofread. (This becomes obvious on a close reading.)
The ingredients are the following.

• Elimination of mixed type
In two papers it is shown that K* groups of finite Morley rank are either of degenerate type, odd type, or even type. The first paper treats the tame case and the second handles the general case. These are in print:
• T. Altinel, A. Borovik, G. Cherlin
Groups of mixed type, J. Alg 192 (1997), 524-571.
• E. Jaligot
Groupes de type mixte, J. Alg 212 (1999), 753-768.
• Strong and weak embedding
This involves two versions of Bender's classification, involving strongly and weakly embedded subgroups. The treatment in the tame case is in
• T. Altinel
Groups of finite Morley rank with strongly embedded subgroups, J. Algebra 180 (1996), 778-807
• T. Altinel, A. Borovik, G. Cherlin
On groups of finite Morley rank with weakly embedded subgroups, J. Algebra 211 (1999), 409-456
The general K* case is treated in
• E. Jaligot
Groupes de rang de Morley fini de type pair avec un sous-groupe faiblement inclus, J. Algebra, 240 (2001), 413-444.
Finally, the L* case is treated in papers by Altinel and Cherlin:
• On groups of Finite Morley Rank of Even Type, T. Altinel and G. Cherlin.
J. Algebra, 264 (2003), 155-185
• Simple L*-groups of even type with strongly embedded subgroups, T. Altinel and G. Cherlin.
J. Algebra 272 (2004), 95-127.
• Simple L*-groups of even type with weakly embedded subgroups, T. Altinel and G. Cherlin
J. Algebra, 286 (2005), 76-106
• Limoncello, T. Altinel and G. Cherlin
J. Algebra, 291 (2005), 371-413.
• Parabolic structure
The weak embedding theorem can be used to eliminate cores of 2-local subgroups, in particular in parabolic subgroups. The next step is to eliminate components of parabolic subgroups.
First one classifies groups with strongly closed abelian subgroups; in the process, one eliminates components of type SL₂.
Then pushing-up and a global C(G,T) theorem are proved, and finally the components are eliminated. The last paper in this series was significantly revised 7/2001.
• T. Altinel, A. Borovik, and G. Cherlin
Groups of finite Morley rank with strongly closed abelian subgroups, J. Algebra 232 (2000), 420-461. (170 K)
• T. Altinel, A. Borovik, and G. Cherlin
Pushing up and C(G,T) in groups of finite Morley rank, J. Algebra, to appear
A. Borovik, G. Cherlin, and L.-J. Corredor
Parabolic 2-local subgroups in groups of finite Morley rank of even type, preprint.
The first of these papers requires substantial elaboration in the L* context, not given in the literature, but included in a book in preparation. The other two pass over rather directly to the L* case.
• Recognition
Depending on the number of minimal parabolic subgroups, one arrives either at a rank 2 amalgam or a situation covered by Niles' theorem.

• The amalgam method
One follows rather closely the method of Stellmacher, concluding with a recognition theorem of Bennett and Shpekterov. This is explained in
• T. Altinel, A. Borovik, and G. Cherlin
K*-groups of even type of finite Morley rank generated by two minimal parabolic subgroups, preprint
This account summarizes the Stellmacher analysis and applies it in the finite Morley rank context. The details of the Stellmacher analysis as it applies in this context will be included in a book in preparation on the L* case.
• Niles' Theorem
The relevant version of Niles' Theorem is found in
• A. Berkman and A. Borovik
An identification theorem for groups of finite Morley rank and even type, preprint, 2000
The L* version of this is
• A generic identification theorem for L*-groups of finite Morley rank A. Berkman, A. Borovik, G. Cherlin, and J. Burdges

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Mixed type

It is a consequence of the Even Type Theorem that there are no simple groups of finite Morley rank and mixed type. This implication has been known for some time, and substantially predates the actual proof of the Even Type Theorem.

More recently it has been shown that groups of degenerate type contain no involutions (by completely different methods). The situation in odd type groups cannot be summarized so simply.

This page copyright A. Altinel, A. Borovik, and G. Cherlin.

This page includes material based upon work supported by the National Science Foundation under successive grants including Grant No 9803417.