有限子群格中区间的尖锐界
Sharp bounds for intervals in finite subgroup lattices
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中文总结 AI 辅助
本文证明了有限群中子群区间内子群数量的尖锐上界，消除了多项式因子，并推广了绝对子群计数定理。
中文摘要 AI 辅助
设 H 是有限群 G 的子群，令 n = [G:H] > 1。若 p 是 n 的最小素因子，我们证明满足 H ≤ K ≤ G 的子群 K 的个数小于 c(p) n^((log_p n)/4)。这里 c(p) 是所有正整数 j 上 (1 - p^(-j))^(-1) 的乘积，再乘以所有整数 z 上 p^(-z^2) 的和。该指数和常数同时是最优的，即使对偶数秩的初等阿贝尔 p-群也是如此。特别地，对任意 n > 1，中间子群的个数小于 7.371968802 n^((log_2 n)/4)。这消除了先前已知最佳相对界中的多项式因子，并将尖锐的绝对子群计数定理推广到任意子群区间。该群论论证是初等的。其主要成分是关于相对生成元组的加权打包不等式，该不等式由陪集空间 G/H 上的逃逸树获得。
英文摘要
Let H be a subgroup of a finite group G, and put n = [G:H] > 1. If p is the least prime divisor of n, we prove that the number of subgroups K with H <= K <= G is less than c(p) n^((log_p n)/4). Here c(p) is the product of (1 - p^(-j))^(-1) over all positive integers j, multiplied by the sum of p^(-z^2) over all integers z. The exponent and the constant are simultaneously optimal, already for elementary abelian p-groups of even rank. In particular, for arbitrary n > 1, the number of intermediate subgroups is less than 7.371968802 n^((log_2 n)/4). This removes the polynomial factor from the best previously known relative bound and extends the sharp absolute subgroup-counting theorem to arbitrary subgroup intervals. The group-theoretic argument is elementary. Its main ingredient is a weighted packing inequality for relative generating tuples, obtained from an escape tree on the coset space G/H.
发表机构
- Beijing Institute of Mathematical Sciences and Applications（北京数学与应用高等研究院）
- Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca（米兰比可卡大学数学与应用系）
机构由 AI 辅助整理，请以论文原文为准。
