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arXiv · 2609.30061

Existence of a Model Companion for Groups of Exponent 3

Yawara Ishida·Ryosuke Mizuno·Kota Takeuchi
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Abstract

In this article, we prove that the theory $T_3$ of groups of exponent $3$ has a model companion. Previous work established the existence of model companions for theories of groups of fixed finite exponent and nilpotency class at most $2$ (Saracino--Wood), and for theories of groups of prime exponent $p$ and nilpotency class at most $c 1$, the theory $T_n$ of groups of exponent $n$ has a model companion if and only if every finitely generated group of exponent $n$ is finite, or equivalently, if and only if the Burnside problem has a positive solution for exponent $n$.

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Yawara Ishida, Ryosuke Mizuno, Kota Takeuchi. 2026-09-24. Existence of a Model Companion for Groups of Exponent 3. https://arxiv.org/abs/2609.30061

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