Subgroup Growth in pro-p Groups
Subgroup Growth in pro-p Groups
复制标题
亲p群体中的子群体增长
DOI：
10.1007/978-1-4612-1380-2_8
复制
发表时间：
2000
影响因子：
0.6
通讯作者：
A. Mann
中科院分区：
文献类型：
--
作者：
A. Mann
设G为任意群。记n（G）为G的指数为n的子群的个数。如果G是一个profinite群，我们只考虑开子群。1如果G是n-生成的，或者作为抽象群或者作为profinite群，则n（G）是有限的。数列a n（G）的研究是由Hurwitz于世纪以几何学为动机而开始的，被称为子群增长，近年来已成为一个活跃的研究领域。让我们提到，例如，一个pro-p群G是p-adic解析的当且仅当它的子群增长是多项式的，即，存在常数C和s，使得对所有的n，an（G）≤ Cn（见下面的第4节和[2]）。p进分析群的这种表征是所有多项式子群增长群（简称PSG群）的表征中的一个工具阶段。子组增长的其他应用在[12]和[4]中找到。
Let G be any group. We write a n (G) for the number of subgroups of G of index n. If G is a profinite group, we consider only open subgroups. 1 If G is finitely generated, either as an abstract or as a profinite group, then a n (G) is finite. The study of the series a n (G), a subject that is known as subgroup growth,was begun by Hurwitz in the 19th century, with geometric motivation, and has become an active area of research in recent years. Let us mention, e.g., that a pro-p Groups G is p-adic analytic if and only if its subgroup growth is polynomial, i.e., there exist constants C and s such that a n (G) ≤ Cns for all n (see Section4 below and [2]). This characterisation of p-adic analytic groups was an instrumental stage in the characterisation of all groups of polynomial subgroup growth (known for short as PSG groups). Other applications of subgroup growth are found in [12] and [4].