Bounds on intermediate subgroups at fixed index

For H≤G of finite index n>1, how many subgroups lie between H and G? The target is a uniform rational upper envelope, including a strict bound c(p)n^((logₚ n)/4) when p is the least prime divisor of n.

Why it matters and what counts as progress

Why it matters: It seeks sharp quantitative control of subgroup intervals and identifies extremal prime-power and elementary-abelian cases.

What counts as progress: A sharper uniform bound on intermediate subgroups at fixed index, or constructions attaining the proposed envelope and its prime-power extremes.

How the problem developed

1996 · Asymptotic bound [1]
Borovik, Pyber, and Shalev Proved an asymptotic upper bound for the total number of subgroups of a finite group; this absolute count preceded sharp estimates for relative subgroup intervals.
2023 · Sharp bound [2]
Fusari and Spiga Removed a polynomial loss from the sharp absolute subgroup-count estimate, obtaining the optimal constant for counting all subgroups of a finite group.
9 Sept 2026 · Theorem [3]
Sébastien Palcoux and Pablo Spiga Proved sharp bounds for intervals in finite subgroup lattices, extending fixed-index subgroup-count estimates to arbitrary subgroup intervals.

AI contribution

GPT-generated bounds count intermediate subgroups

For H≤G of finite index n>1, Palcoux’s manuscript and formal development give a finite rational upper envelope for the number of intermediate subgroups and a strict bound c(p)n^((logₚn)/4), where p is the least prime divisor of n. Selected results also address prime-power and elementary-abelian sharpness.

AI involvementAI-autonomous

ImportanceModest

Technical verificationFormal (Lean 4.33.0 kernel with Mathlib, Comparator and NanoDa)

Human understandingFew