Submission date: 8 January 2020

Abstract:

The convex subsets of a group appeared in POIZAT 2018, motivated by FRECON 2018 whose proof by contradiction consists in the construction of a convex set of dimension two (“a plane”), and then in showing that such a plane cannot exist. In a group of finite Morley rank without involutions, to a definable convex subset are associated symmetries and translations, that we undertake here to study in the abstract, without mentionning a group envelopping them. For this reason we introduce axiomatically a certain kind of structures that we call symmetrons. Glauberman's Z*-Theorem allows to elucidate completely the finite symmetrons: each of them is isomorphic to the set of symmetries associated to a convex subset of a finite group without involutions, which is far from being uniquely determined. In fact, there exist nonisomorphic finite groups which have the same symmetries, and also finite symmetrons which are not isomorphic to the symmetries of a group.
The situation is not so clear in the case of symmetrons of finite Morley rank, or even algebraic, which are the main objects of study of this paper. But in spite of the fact that a symmetron be a structure much weaker that a group, we can extend to symmetrons some well-known results concerning groups of finite Morley rank: chain condition, decomposition into connected components, characterisation of the generic definable subsets, elliptic generation, etc.. Moreover, assuming the Algebricity Conjecture, we generalize Glauberman's Theorem to the finite Morley rank context.

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