论文标题
Frattini注射率和最大Pro-P $ GALOIS组
Frattini-injectivity and Maximal pro-$p$ Galois groups
论文作者
论文摘要
如果$ g $有独特的有限生成的亚组，我们将其称为Pro-P $ G $ G $ G $ frattini注入。本文是针对系统研究Frattini Injextive Pro-P $组（以及其他几个相关概念）的最初努力。最值得注意的是，我们对$ p $ -ADIC分析和可解决的Frattini Injextive Pro-P $组组进行了分类，并描述了frattini Injuntive Pro-P $组的普通Abelian亚组的晶格。 我们证明，每个最大pro- $ p $ p $ galois集团都包含一个原始的$ p $ p $ th根（并且还包含$ \ sqrt {-1} $，如果$ p = 2 $）是frattini indentive。此外，我们表明，最大奖金$ p $ galois组的许多实质性结果实际上是弗拉蒂尼注射率的后果。例如，$ p $ - adic分析或可解决的pro- $ p $组是frattini indextive的，并且只有当它可以实现为最大pro-p $ p $ p $ p $ p $ p $ p $ galois集团，其中包含一个原始的$ p $ p $ p $ th root of unity（并且还包含$ \ sqrt {-1} $ p = $ p = 2 $）;每个弗拉蒂尼注射pro-p $组都包含一个唯一的最大阿贝利安普通亚组。
We call a pro-$p$ group $G$ Frattini-injective if distinct finitely generated subgroups of $G$ have distinct Frattinis. This paper is an initial effort toward a systematic study of Frattini-injective pro-$p$ groups (and several other related concepts). Most notably, we classify the $p$-adic analytic and the solvable Frattini-injective pro-$p$ groups, and we describe the lattice of normal abelian subgroups of a Frattini-injective pro-$p$ group. We prove that every maximal pro-$p$ Galois group of a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is Frattini-injective. In addition, we show that many substantial results on maximal pro-$p$ Galois groups are in fact consequences of Frattini-injectivity. For instance, a $p$-adic analytic or solvable pro-$p$ group is Frattini-injective if and only if it can be realized as a maximal pro-$p$ Galois group of a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$); and every Frattini-injective pro-$p$ group contains a unique maximal abelian normal subgroup.
