Abstract
1. In an Abelian group, a module, or more generally a one-based group H, the only definable groups are the obvious ones: if G is interpretable in H, then it has a definable subgroup of finite index which is definably isomorphic to a quotient A/B, where A and B are definable subgroups of a Cartesian power of H. 2. In such a group the introduction of those quotient groups weakly eliminates imaginary elements. More generally, for a stable theory the existence of canonical real bases for complete types implies the elimination of imaginary elements. 3. A group which is interpretable in a one-based structure is one-based. The property of being one-based is preserved by interpretation for theories of finite rank but not in general.
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Translated from French.
Translated from Algebra i Logika, No. 3, pp. 368–378, May–June, 1990.
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Evans, D., Pillay, A. & Poizat, B. A group in a group. Algebra and Logic 29, 244–252 (1990). https://doi.org/10.1007/BF01979940
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DOI: https://doi.org/10.1007/BF01979940