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Weakly normal groups. (English) Zbl 0636.03028
Logic colloq. ’85, Proc. Colloq., Orsay/France 1985, Stud. Logic Found. Math. 122, 233-244 (1987).
[For the entire collection see Zbl 0611.00002.]
Roughly a theory T is said to be weakly normal (or 1-based) if every infinite set of pairwise distinct conjugates of a given definable set X has empty intersection; a group G is weakly normal if its theory is. The authors show that a group G is weakly normal if and only if, for each n, every definable \(X\subseteq G\) n is a boolean combination of cosets of definable (over acl(\(\emptyset))\) subgroups of G n. Moreover such a group is abelian-by-finite. The key lemma asserts that, if G is weakly normal and \(| T\) \(+|\)-saturated, then each type over G can be translated into the generic type of a definable subgroup of G.
The result is a useful fact about general 1-based theories, at least in the superstable context, since, by a recent result of Hrushovski, if q is a nontrivial regular type of such a theory T, then T interprets a group G such that q is domination-equivalent to the generic type of a subgroup of G.
Reviewer: Ch.Berline

03C45 Classification theory, stability, and related concepts in model theory
03C60 Model-theoretic algebra