Generation of finite groups from subgroups of coprime index

Abstract

Let $d(G)$ denote the least size of a generating set of a finite group $G$. We prove that if $G$ has a family $\mathcal H$ of subgroups such that $d(H)\leq d$ for every $H\in\mathcal H$ and $\gcd\{\lvert G:H\rvert:H\in\mathcal H\}=1$, then $d(G)\leq d+1$. This gives an affirmative answer to Kourovka Problem 21.87. The proof reduces a minimal counterexample to a critical crown-based power with nonabelian socle. An exact crown multiplicity formula and a uniform lower bound for conditional generation give a lower bound for the number of crown factors. A subgroup containing a Sylow $2$-subgroup gives the contradictory upper bound, via a pointwise centralizer estimate for Sylow $2$-subgroups of finite simple groups.

Disclosure

“al generation, and generation of simple groups depend on the classification of finite simple groups. A literature search conducted before submission found no earlier paper asserting this conclusion, but this does not establish priority. OpenAI Codex was used for literature organization, proof stress-testing, and editorial assistance. The author is responsible for all content and mathematical claims in this manuscript. References 1. Timothy C”

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Proof ideas or individual proof-step assistance

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8

Verified

2026-08-23T05:11:24.864971Z

Structural counts