Combinatorial group theory for pro-p groups
Combinatorial group theory for pro-p groups
复制标题
Pro-p 群的组合群论
DOI：
10.1016/0022-4049(82)90086-x
复制
发表时间：
1982
影响因子：
0.8
通讯作者：
A. Lubotzky
中科院分区：
文献类型：
--
作者：
A. Lubotzky
虽然亲有限群范畴是有限群范畴的自然延伸，但它具有更丰富的结构，因为它具有有限情况下不存在的范畴对象和概念;例如投射群和自由积。这些概念在扩展范畴中的存在导致了组合群论中通常概念的定义，例如自由群和通过生成元和关系定义群。在不同领域的主题导致亲p组的特殊考虑：塔的问题（参见。[21])，p-adic域上的伽罗瓦理论和Demuskin群（cf. [20])，通过上同调的方式解释生成元和关系（cf. [19，2 l]），幂零群理论（cf. [ll])然而，没有系统的理论（但见[6]）。本文的目的是开始发展我们称之为pro-p群的组合群论，尽管组合工具在这里似乎没有用。组合群论的基础书籍[16]和[15]都开始从自由群，它们的子群和它们的自同构开始。因此，我们研究这些方面的亲p组。在总结了自由群和自由积的基本（和大多数众所周知的）性质之后，我们在第3节中证明了一些类似于Hall，Greenberg和Howson关于自由群的n-生成子群的定理的结果。在第五节中，我们描述了n-生成的自由pro-p群的自同构群，并得到了一个推论：与离散情形相反，两个生成元上的自由pro-p群有一个外自同构平凡地作用于换位子商。亲p群的Frattini子群在我们的工作中起着核心作用;它的基本性质在第1节中总结。这个概念，其重要性pro-p群首先注意到由Gruenberg [7]，使我们能够将组合群论的概念与群论的概念（例如见3.1）联系起来，因此我们可以用非常基本的群论方法代替组合方法。还要注意，我们的方法是免费的上同调。
Although the category of pro-finite groups forms a natural extension of the category of finite groups, it carries a richer structure in that it has categorical objects and notions which do not exist in the finite case; eg projective groups and free product. The existence of such notions in the extended category leads to the definition of the usual notions of combinatorial group theory, such as free groups and defining a group by generators and relations. Topics in various fields lead to a special consideration of pro-p groups: the tower problem (cf.[21]), Galois theory over p-adic fields and Demuskin groups (cf.[20]), the interpretation of generators and relations by means of cohomology (cf.[19, 2 l]), the theory of nilpotent groups (cf.[ll]) etc. Nevertheless, there is no systematic theory (but see [6]). The aim of this paper is to begin to develop what we call combinatorial group theory for pro-p groups, although combinatorial tools do not seem to be useful here.The fundamental books on combinatorial group theory,[16] and [15] both begin with free groups, their subgroups and their automorphisms. Accordingly, we study these aspects of pro-p groups. After summarizing (in Section 2) the basic (and mostly well-known) properties of free groups and free products, we prove in Section 3 some results analogous to the theorems of Hall, Greenberg and Howson about finitely generated subgroups of free groups. In Section 5, we describe the automorphism group of finitely generated free pro-p groups, and obtain as a corollary that, contrary to the discrete case, a free pro-p group on two generators has an outer automorphism acting trivially on the commutator quotient. A central role in our work is played by the Frattini subgroup of a pro-p group; its basic properties are summarized in Section 1. This notion, whose importance to pro-p groups was first noted by Gruenberg [7], enables us to relate combinatorial group theoretic notions to group theoretic ones (see for example 3.1) and so we can replace the combinatorial methods by quite elementary group theoretic methods. Note also that our methods are free from cohomology.