发表机构
A.I. Systems Research Institute Co., Ltd.; Institute of Mathematics, University of Tsukuba（A.I. Systems Research Institute Co., Ltd.; 筑波大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明指数为3的群的理论具有模型伴生，验证了关于指数n的群理论模型伴生存在性猜想中n=3的情形，该猜想与Burnside问题的正解相关。
AI 中文摘要
本文证明了指数为3的群的理论$T_3$具有模型伴生。先前的工作确立了固定有限指数且幂零类至多为2的群的理论（Saracino--Wood）以及素数指数$p$且幂零类至多$c<p$的群的理论（Maier）的模型伴生的存在性。这些结果不涵盖$T_3$，因为指数为3的群可能具有幂零类3。我们的定理验证了第三作者提出的猜想中$n=3$的情形，该猜想指出：对于每个整数$n>1$，指数为$n$的群的理论$T_n$具有模型伴生当且仅当每个有限生成的指数为$n$的群是有限的，等价地，当且仅当Burnside问题对指数$n$有正解。
英文摘要
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 $n$.
