Sharp bounds for intervals in finite subgroup lattices
Abstract
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.
Related
- M-subgroups and normal-subgroup lattices of finite groups
- On a finite group in which every non-supersolvable maximal subgroup is a TI-subgroup
- Convex Bernstein limits and sharp recovery on expanding intervals
- On a finite group in which every maximal subgroup of order divisible by p containing the normalizer of some Sylow subgroup is nilpotent
- Improved Weyl bounds on short intervals
BibTeX
@misc{palcoux2026sharp,
title = {{Sharp bounds for intervals in finite subgroup lattices}},
author = {Sebastien Palcoux and Pablo Spiga},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10343},
eprint = {2609.10343},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10343},
}
RIS
TY - UNPB
TI - Sharp bounds for intervals in finite subgroup lattices
AU - Sebastien Palcoux
AU - Pablo Spiga
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10343
UR - https://arxiv.org/abs/2609.10343
AB - 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.
ER -