Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a continuous representation $θ\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ such that every open subgroup $H$ of $G$, together with the restriction $θ\vert_H$, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-$p$ group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-$p$ Galois groups of fields, and that if a 1-smooth pro-$p$ group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-$p$ Galois groups of fields. Finally we ask whether 1-smooth pro-$p$ groups satisfy a "Tits' alternative".
Università di Milano-Bicocca（米兰比可卡大学）；Université Marie et Louis Pasteur（玛丽和路易·巴斯德大学）；Harbin Institute of Technology（哈尔滨工业大学）；Korea National University of Education（韩国国立教育大学）
Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions
给定一个有限群$ G $，研究了$ G $的置换群的正则子群，它们出现在两类代数对象的分类理论中：具有乘法群同构于$ G $的斜左括号和具有伽罗瓦群同构于$ G $的域的伽罗瓦扩张所允许的Hopf-Galois结构.研究了两个这样的子群何时产生同构的斜左括号或同构的霍普夫代数的Hopf-Galois结构。特别地，我们证明了在某些情况下，给出一个Hopf-Galois结构的霍普夫代数的同构类由相应的斜左支撑决定。我们调查这些问题的背景下，各种现有的结构在文献中。作为我们的结果的应用，我们分类的同构不同的霍普夫代数，给出了度$ pq $的伽罗瓦扩张上的Hopf-Galois结构的$ p>q $素数。
