发表机构
Eastern Institute of Technology（东方理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明群可由含单位元的对称生成子集恢复当且仅当它是平凡群或2、3阶循环群，通过局部映射延拓定理等推导，构造了至多5元域上的障碍，验证了P. A. Smith的猜想。
AI 中文摘要
对于群中包含单位元的对称子集，其通用局部表示记录了该子集内可见的乘法关系。我们证明，一个群可由每个此类生成子集恢复，当且仅当它是平凡群或2阶、3阶循环群，这符合P. A. Smith的猜想。核心步骤是一个精确完备化定理：当且仅当目标群的每个元素都有平方根时，到目标群的局部映射总能延拓到生成域。通过将任意群嵌入具有满射平方映射的群中，我们推导出每个部分局部表示都可嵌入生成局部表示。这给出子群继承性，并将分类问题简化为有限生成性。我们还在至多含5个元素的域上，于一个固定的可数亚阿贝尔目标群中构造了障碍，该界即使允许目标群变化仍是紧的。
英文摘要
For a symmetric subset of a group containing the identity, its universal local presentation records the multiplication relations visible within the subset. We prove that a group is recovered from every such generating subset if and only if it is trivial or cyclic of order two or three, as conjectured by P. A. Smith. The main step is an exact completion theorem: local maps into a target group always extend to generating domains if and only if every target element has a square root. By embedding an arbitrary group in a group with surjective squaring, we deduce that every partial local presentation embeds in a generating one. This gives subgroup inheritance and reduces the classification to finite generation. We also construct obstructions in one fixed countable metabelian target on domains of at most five elements. This bound is sharp even when the target is allowed to vary.
Comments9 pages, 0 figures
