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Omega-categorical pseudofinite groups

Dugald Macpherson, Katrin Tent

Abstract

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.

Omega-categorical pseudofinite groups

Abstract

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.
Paper Structure (3 sections, 16 theorems, 9 equations)

This paper contains 3 sections, 16 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Saracino-Wood groups and variants
  3. Further observations

Key Result

Proposition 1.2

Conjecture mainq holds if every $\omega$-categorical pseudofinite $p$-group of class at most 2 is FAF.

Theorems & Definitions (32)

  • Conjecture 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • ...and 22 more