Omega-categorical pseudofinite groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929289511829504 |
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| author | Macpherson, Dugald Tent, Katrin |
| author_facet | Macpherson, Dugald Tent, Katrin |
| contents | We explore the interplay between omega-categoricity and pseudofiniteness for groups, conjecturing that omega-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of omega-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the (omega-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an omega-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on omega-categorical groups over 50 years. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Omega-categorical pseudofinite groups Macpherson, Dugald Tent, Katrin Logic Group Theory 03C60 (Primary), 20A15 (Secondary) We explore the interplay between omega-categoricity and pseudofiniteness for groups, conjecturing that omega-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of omega-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the (omega-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an omega-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on omega-categorical groups over 50 years. |
| title | Omega-categorical pseudofinite groups |
| topic | Logic Group Theory 03C60 (Primary), 20A15 (Secondary) |
| url | https://arxiv.org/abs/2403.17684 |