Lascar and Morley ranks differ in differentially closed fields

Ehud Hrushovski; Thomas Scanlon

doi:10.2307/2586629

Lascar and Morley ranks differ in differentially closed fields

Journal of Symbolic Logic ◽

10.2307/2586629 ◽

1999 ◽

Vol 64 (3) ◽

pp. 1280-1284 ◽

Cited By ~ 2

Author(s):

Ehud Hrushovski ◽

Thomas Scanlon

Keyword(s):

Fundamental Issue ◽

Morley Rank ◽

Closed Fields ◽

The Family ◽

Finite Morley Rank

We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. Section 1 consists of some general lemmas relating the above issues. Section 2 points out a family of sets of finite Morley rank, whose Morley rank exhibits discontinuous upward jumps. To make the base of the family itself have finite Morley rank, we use a theorem of Buium.

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Schur-Zassenhaus theorem revisited

Journal of Symbolic Logic ◽

10.2307/2275265 ◽

1994 ◽

Vol 59 (1) ◽

pp. 283-291 ◽

Cited By ~ 11

Author(s):

Alexandre V. Borovik ◽

Ali Nesin

Keyword(s):

Solvable Groups ◽

Hall Subgroup ◽

Unipotent Radical ◽

Morley Rank ◽

Stable Group ◽

Similar Question ◽

Closed Fields ◽

Finite Morley Rank ◽

Divisible Subgroup ◽

Algebraically Closed Fields

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.Theorem 2. Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.This result has been proved in [1] with the additional assumption that G is connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.The second author has shown in [11] that a nilpotent ω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus.

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