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 $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ such that every open subgroup H of G, together with the restriction $\theta \vert _H$ , 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.”

设p为素数。如果一个 pro - p群G可以被赋予一个连续表示 $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ 使得每个开子群G的H与限制 $\theta \vert _H$ 一起满足 Hilbert 90 的正式版本。我们证明每个 1-光滑的 pro- p群都包含一个唯一的最大闭阿贝尔正规子群，与结果类似Engler 和 Koenigsmann 关于最大 pro- p Galois 场群，并且如果一个 1-smooth pro- p群是可解的，那么它是局部一致强大的，类似于 Ware 对最大 pro- p伽罗瓦域群的结果。最后，我们询问 1-smooth pro- p组是否满足“Tits 的替代方案”。