pro-π群的秩与维度的有限可公理化
Finite axiomatizability of the rank and the dimension of a pro-$π$ group
AI总结：
本文证明了在群的一阶语言中存在单一语句，能够精确识别有界秩pro-π群的秩及其Sylow子群的秩和维度。
摘要：
profinite群G的Prüfer秩rk(G)是G的所有开子群H的最小生成元数d(H)的上确界。已知对于任意给定的素数p，一个profinite群G容许p-adic解析群结构当且仅当G实质上是有限秩的pro-p群。p-adic解析profinite群G的维度dim G是G作为p-adic流形的解析维度；已知dim G与G的任何一致强大开pro-p子群U的秩rk(U)一致。设π为有限的素数集合，设r∈N且设r=(r_p)_{p∈π}, d=(d_p)_{p∈π}为{0,1,...,r}中的元组。我们证明在群的一阶语言中存在单一语句σ_{π,r,r,d}，使得对于每个pro-π群G，以下等价：(i) σ_{π,r,r,d}在群G中成立，即G⊨σ_{π,r,r,d}；(ii) G的秩为r，且对于每个p∈π，G的Sylow pro-p子群的秩为r_p，维度为d_p。粗略地说，这表明对于有界秩的pro-π群G，G的精确秩以及G的Sylow子群的秩和维度可以通过群的一阶语言中的单一语句来识别。
英文摘要：
The Prüfer rank $\mathrm{rk}(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $\mathrm{d}(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits the structure of a $p$-adic analytic group if and only if $G$ is virtually a pro-$p$ group of finite rank. The dimension $\dim G$ of a $p$-adic analytic profinite group $G$ is the analytic dimension of $G$ as a $p$-adic manifold; it is known that $\dim G$ coincides with the rank $\mathrm{rk}(U)$ of any uniformly powerful open pro-$p$ subgroup $U$ of $G$. Let $π$ be a finite set of primes, let $r \in \mathbb{N}$ and let $\mathbf{r} = (r_p)_{p \in π}, \mathbf{d} = (d_p)_{p \in π}$ be tuples in $\{0, 1, \ldots,r\}$. We show that there is a single sentence $σ_{π,r,\mathbf{r},\mathbf{d}}$ in the first-order language of groups such that for every pro-$π$ group $G$ the following are equivalent: (i) $σ_{π,r,\mathbf{r},\mathbf{d}}$ holds true in the group $G$, that is, $G \models σ_{π,r,\mathbf{r},\mathbf{d}}$; (ii) $G$ has rank $r$ and, for each $p \in π$, the Sylow pro-$p$ subgroups of $G$ have rank $r_p$ and dimension $d_p$. Loosely speaking, this shows that, for a pro-$π$ group $G$ of bounded rank, the precise rank of $G$ as well as the ranks and dimensions of the Sylow subgroups of $G$ can be recognized by a single sentence in the first-order language of groups.
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