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Preprint Number 1717
1717. Bruno Poizat Symétries et translations, principalement dans les groupes de rang de Morley fini sans involutions Email: 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. Mathematics Subject Classification: Keywords and phrases: Full text: pdf.

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