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OMEGA-CATEGORICAL PSEUDOFINITE GROUPS

Part of: Foundations Special aspects of infinite or finite groups Model theory

Published online by Cambridge University Press:  15 January 2025

DUGALD MACPHERSON Open the ORCID record for DUGALD MACPHERSON [Opens in a new window]  and
KATRIN TENT
DUGALD MACPHERSON*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS LEEDS UK
KATRIN TENT
Affiliation:
MATHEMATISCHES INSTITUT UNIVERSITY OF MÜNSTER MÜNSTER GERMANY E-mail: tent@wwu.de
*
E-mail: h.d.macpherson@leeds.ac.uk
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Abstract

We explore the interplay between $\omega $-categoricity and pseudofiniteness for groups, and we conjecture 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.

Keywords

omega-categoricalpseudofinitenilpotent

MSC classification

Primary: 03C60: Model-theoretic algebra 20A15: Applications of logic to group theory 20F50: Periodic groups; locally finite groups
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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