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 $d(H)\le d$ for every $H\in \mathcal{H}$ and $\gcd \{|G:H|:H\in \mathcal{H}\}=1$, then $d(G)\le d+1$. 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.

7 pages. Supporting source and proof-audit materials are available at https://github.com/RichieSater/kourovka-21-87 and archived at https://doi.org/10.5281/zenodo.21893748