## A group in a group.(English. French original)Zbl 0744.03032

Algebra Logic 29, No. 3, 244-252 (1990); translation from Algebra Logika 29, No. 3, 368-378 (1990).
A stable theory $$T$$ is said to be weakly normal if, for every model $$M$$ of $$T$$, every subset $$A$$ of $$M$$ and every element $$b$$ in $$M$$, the canonical basis of $$t(b/A)$$ is algebraic over $$A\cup b$$. It is known that, if $$T$$ is a superstable theory of finite rank, $$T$$ is weakly normal if and only if all rank-one types have a locally modular geometry.
The main result of this paper is the following theorem: assume that $$T$$ is a weakly normal theory of groups (i.e. $$T$$ is weakly normal and there is a definable group structure over the universe of any model of $$T$$); then, if $$H$$ is a group which is interpretable over a model $$G$$ of $$T$$, then $$H$$ has a subgroup of finite index which is definably isomorphic to $$A/B$$, where $$A$$ and $$B$$ are definable subgroups of some cartesian power of $$G$$. As a corollary, the authors get the fact that $$(\mathbb{Z}/4\mathbb{Z})^ \omega$$ cannot be coordinated by its subgroup $$H$$ of elements of order 2 (isomorphic to $$(\mathbb{Z}/2\mathbb{Z})^ \omega)$$, which was the question motivating this research.
Reviewer: D.Lascar (Paris)

### MSC:

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

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