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By R. Göbel, L. Lady, A. Mader

ISBN-10: 3540123350

ISBN-13: 9783540123354

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D Note that G° is 0-definable because (i) G° is definable and (ii) G° can be described without reference to any parameters. 3. Let G be co-stable, pe Si(G) is said to be a generic type of G if RM(p) = RM(G) (= RM("x = x") in Th(G)). 4. (G co-stable) (i) There are only finitely many generic pe Si(G) (and there is at least one. (ii) if p is generic so is p"1 (by *). (iii) G acts (by left or right translation) on the generic types of Si(G) (by *). 5. G acts (by left or right translation) transitively and definably on the set of generics of Si(G).

R let Vi = V(pi) = smallest Zariski closed set in pi. Clearly Vi is irreducible (as K is a model) and Vi ci V. By quantifier elimination pi is "determined" by Vj. So if f is a k-automorphism of K then for any i, f(VO = Vj for some j. Thus each V{ has only finitely many conjugates over k. ,Vis, be the k-conjugates of Vi. U Vis is defined over k, is in p = tp(a/k) and so equals V. UVr = V and that the Vi are the irreducible components of V. This suffices to prove the proposition. 21. (Again k perfect).

G. A is a normal subgroup of G and G acts by conjugation). We call X d A Ginvariant if X is fixed setwise by G. 13: Let X c: A be G-invariant. Then X is indecomposable if and only if for every definable G-invariant subgroup H of A we have IX/HI = 1 or oo. Proof: Let H be an arbitrary definable subgroup of A. ,sne G. n H*" is G-invariant. 14. Let G, A be (infinite) definable Abelian groups in a structure M of finite Morley rank and assume that G acts definably and faithfully on A. Suppose moreover that A has no infinite proper G-invariant subgroup.

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Abelian Group Theory. Proc. conf. Honolulu, 1983 by R. Göbel, L. Lady, A. Mader


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