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OMEGA-CATEGORICAL PSEUDOFINITE GROUPS
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 15 January 2025, pp. 1-14
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- Article
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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.
