Counting subgroups of a finite group containing a prescribed subgroup

TL;DR

Let $R$ be a finite group and $T\le R$. The paper proves an explicit upper bound on the number of subgroups of $R$ that contain $T$, namely $|\mathrm{Sub}(R,T)|\le 7.3722\cdot [R:T]^{\frac{\log_2[R:T]}{4}+1.8919}$. The proof reduces to analyzing $p$-groups via a decomposition over distinct primes, bounding $|\mathrm{Sub}(P,T)|$ in a $p$-group by Gaussian binomial coefficients and then combining the data across primes using Sylow theory and double coset counting to obtain a global bound; an intricate arithmetic step refines the exponent with a constant $\alpha=1.8919$. A permutation-group corollary states that a transitive group of degree $n$ has at most $7.3722\cdot n^{\frac{\log_2 n}{4}+1.8919}$ systems of imprimitivity. The paper also discusses potential tightening to replace $\log_2 [R:T]$ by the number of prime divisors $\lambda([R:T])$ and outlines the steps toward that refinement.

Abstract

Let $R$ be a finite group, and let $T$ be a subgroup of $R$. We show that there are at most \[ 7.3722[R:T]^{\frac{\log_2[R:T]}{4}+1.8919} \] subgroups of $R$ containing $T$.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Conjecture 1.3
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3