##
**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.

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 |

### Keywords:

stable theory; weakly normal theory of groups; definable group structure; definable subgroups; cartesian power
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\textit{D. Evans} et al., Algebra Logic 29, No. 3, 244--252 (1990; Zbl 0744.03032); translation from Algebra Logika 29, No. 3, 368--378 (1990)

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### References:

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[4] | B. Poizat, ”A propos de groupes stables,” in: Logic Colloquium ’85, North Holland (1987), pp. 245–265. · Zbl 0626.03025 |

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