Existence of a Model Companion for Groups of Exponent 3
Abstract
In this article, we prove that the theory T3 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<p (Maier). These results do not cover T3, since groups of exponent $3$ may have nilpotency class $3$. Our theorem verifies the n=3 case of a conjecture proposed by the third author that, for each integer n>1, the theory Tn 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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BibTeX
@misc{ishida2026existence,
title = {{Existence of a Model Companion for Groups of Exponent 3}},
author = {Yawara Ishida and Ryosuke Mizuno and Kota Takeuchi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30061},
eprint = {2609.30061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30061},
}
RIS
TY - UNPB
TI - Existence of a Model Companion for Groups of Exponent 3
AU - Yawara Ishida
AU - Ryosuke Mizuno
AU - Kota Takeuchi
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30061
UR - https://arxiv.org/abs/2609.30061
AB - 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<p (Maier). These results do not cover T_3, since groups of exponent $3$ may have nilpotency class $3$. Our theorem verifies the n=3 case of a conjecture proposed by the third author that, for each integer n>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.
ER -