A new trichotomy theorem for groups of finite Morley rank

Abstract

There is a longstanding conjecture, of Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Here we will conclude that a simple K * -group of finite Morley rank and odd type either has normal rank of at most 2, or else is an algebraic group over an algebraically closed field of characteristic not 2. To this end, it suffices to produce a proper 2-generated core in groups with Prüfer rank 2 and normal rank at least 3, which is what is proved here. Our final conclusion constrains the Sylow 2-subgroups available to a minimal counterexample and, finally, proves the trichotomy theorem in the nontame context.

Signalizers and balance in groups of finite Morley rank

“…Borovik's earlier unpublished work on the analysis of Lie rank two components [Bor03] has heavily influenced this final result, although partial balance has given these results a more technical flavor.…”

“…This section also contains a version of Asar's theorem (Theorem 2.12) which states that ẼX = ∅, as well as a criterion for ẼX = G (Theorem 2.18). Borovik's earlier unpublished work on the analysis of Lie rank two components [Bor03] has heavily influenced this final result, although partial balance has given these results a more technical flavor.…”

Groups of finite Morley rank with a generically sharply multiply transitive action

“…For a group G of odd type, the crucial parameter is the Prüfer 2-rank, pr(G), that is, the number of copies of Z 2 ∞ in the direct sum decomposition T = Z 2 ∞ × · · · × Z 2 ∞ of a maximal divisible 2-subgroup T of G. The present state of affairs is stated in the following theorem, which summarises a series of works by Altınel, Berkman, Borovik, Burdges, Cherlin, Deloro, Frécon, and Jaligot [1,3,4,6,7,8,9,10,14,15,16,17,18,19,20,21,23,25,26,28,33,34,35,36] and reduces the classification of groups of odd type to a number of "small" configurations. Fact 1.2.…”

“…The present state of affairs is stated in the following theorem, which summarises a series of works by Altınel, Berkman, Borovik, Burdges, Cherlin, Deloro, Frécon, and Jaligot [1,3,4,6,7,8,9,10,14,15,16,17,18,19,20,21,23,25,26,28,33,34,35,36] and reduces the classification of groups of odd type to a number of "small" configurations. Fact 1.2.…”