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Redidual properties in supersimple groups. (Propriétés résiduelles dans les groupes supersimples.) (French. English summary) Zbl 1222.03038

Let \(C\) be a pseudo-variety, that is, a class of groups closed under subgroups, factor groups and finite products. Call a group \(G\) residually \(C\) if, for every element \(g \neq 1_G\) of \(G\), there is a normal subgroup \(N\) of \(G\) such that \(G/N \in C\) and \(g \not \in N\). The author shows that, if \(G\) is a supersimple residually \(C\) group, then there is a normal series of definable subgroups of \(G\), \(G = G_0 \supseteq G_1 \supseteq \ldots \supseteq G_n\), such that, for every \(i < n\), \(G_i/G_{i+1} \in C\) and \(G_n\) is nilpotent. This extends in the semisimple setting and strenghtens previous results by Ould Houcine for superstable residually \(C\) groups.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
20A15 Applications of logic to group theory
20E26 Residual properties and generalizations; residually finite groups
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References:

[1] Simple theories (2000)
[2] DOI: 10.1002/malq.200610023 · Zbl 1109.03028 · doi:10.1002/malq.200610023
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