论文标题
frattini-pro-$ p $ groups的抗frattini直接产品
Frattini-resistant direct products of pro-$p$ groups
论文作者
论文摘要
如果功能$ h \ mapstoφ（h）$，从所有$ g $本身的所有封闭子组的poset中，pro-p $ g $ g $被称为强烈耐药的，则称为poset嵌入。抗frattini-po $ p $群体自然而然地出现在Galois理论中。实际上，在包含一个原始的$ p $ p $ th根部的字段上的每个最大pro-p $ p $ galois组（如果$ p = 2 $）具有强烈的耐药性。令$ g_1 $和$ g_2 $为非平凡的pro- $ p $组。我们证明，$ g_1 \ times g_2 $在且仅当直接因素之一$ g_1 $或$ g_2 $之一是无扭转的阿贝利安（Abelian）时，并且另一个具有其所有封闭子组的财产，而其所有封闭子组都具有无扭力的亚替体化。作为推论，我们获得了Koenigsmann在最大值$ $ P $ GALOIS组上的群体理论证明，该基础将非平凡的分解作为直接产品。此外，我们举例说明了一个不强烈耐frattini的组的示例，但其属性是其frattini功能定义了所有拓扑上生成的亚组的POSET的订单自我变为。
A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \mapsto Φ(H)$, from the poset of all closed subgroups of $G$ into itself, is a poset embedding. Frattini-resistant pro-$p$ groups appear naturally in Galois theory. Indeed, every maximal pro-$p$ Galois group over a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is strongly Frattini-resistant. Let $G_1$ and $G_2$ be non-trivial pro-$p$ groups. We prove that $G_1 \times G_2$ is strongly Frattini-resistant if and only if one of the direct factors $G_1$ or $G_2$ is torsion-free abelian and the other one has the property that all of its closed subgroups have torsion-free abelianization. As a corollary we obtain a group theoretic proof of a result of Koenigsmann on maximal pro-$p$ Galois groups that admit a non-trivial decomposition as a direct product. In addition, we give an example of a group that is not strongly Frattini-resistant, but has the property that its Frattini-function defines an order self-embedding of the poset of all topologically finitely generated subgroups.
