稠密局部极小群
Densely locally minimal groups
摘要：
研究了所有稠密子群（局部）极小的局部紧群.我们称这样的群为稠密（局部）极小群。在1972年普罗达诺夫证明，无限紧阿贝尔群的所有小组最小正是集团$\mathbb Z_p$的$p$-进整数。在[31]中，我们在几个层次上将Prodanov定理推广到非阿贝尔情形。本文主要研究稠密（局部）极小阿贝尔群。证明了拓扑交换群$G$是紧的或连通的局部紧的，则$G$是稠密局部极小的当且仅当$G$是李群或对某个素数$p$有一个开子群同构于$\mathbb Z_p$.这应该与[9]的主要结果相比较。我们的定理C提供了普罗达诺夫定理的另一个推广：一个无限局部紧群是稠密极小的当且仅当它同构于$\mathbb Z_p$。与此相反，我们证明了存在一个稠密的极小的，紧的，两步幂零群，既不是李群，也没有一个开子群同构于$\mathbb Z_p$。
英文摘要：
We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups $\mathbb Z_p$ of $p$-adic integers. In [31], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group $G$ is either compact or connected locally compact, then $G$ is densely locally minimal if and only if $G$ either is a Lie group or has an open subgroup isomorphic to $\mathbb Z_p$ for some prime $p$. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to $\mathbb Z_p$. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to $\mathbb Z_p$.
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