强简洁性在 profinite 亚阿贝尔群中的应用
Strong conciseness in profinite metabelian groups
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中文总结 AI 辅助
该研究证明每个群字在 profinite 亚阿贝尔群类中具有强简洁性，并借助新方法给出了 profinite 幂零群类中对应结论的另一种证明。
中文摘要 AI 辅助
群字 $w$ 在 profinite 群类 $\mathcal{C}$ 中被称为强简洁的，若对每个 $G \in \mathcal{C}$，当 $w$ 在 $G$ 中的取值少于 $2^{\aleph_0}$ 个时，其对应的 verbal 子群 $w(G)$ 有限。利用 Passi 引入的多项式映射概念，我们证明每个群字在 profinite 亚阿贝尔群类中均为强简洁的，还通过该新方法给出了“每个群字在 profinite 幂零群类中强简洁”这一结论的另一种证明。
英文摘要
A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G \in \mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Using the notion of polynomial mappings introduced by Passi, we establish that every group word is strongly concise in the class of profinite metabelian groups. With this new approach, we also give an alternative proof for the fact that every group word is strongly concise in the class of profinite nilpotent groups.
