Generation of finite groups from subgroups of coprime index

Abstract

Version 1.1.3 of the preprint and its supporting source records answer Kourovka Notebook Problem 21.87. Let d(G) be the least size of a generating set of a finite group G. If G has a nonempty family of subgroups, each generated by at most d elements, whose indices have greatest common divisor one, then d(G) is at most d+1. The bound is sharp. The proof reduces a least counterexample to a critical nonabelian crown. Conditional generation gives a lower count with factor 1/2. The pointwise Sylow-centralizer estimate proved in the manuscript and a crown-fiber orbit-packing argument give an upper count with factor 1/4, producing a strict contradiction. No computation is performed in the manuscript. The published external theorems are assumed as stated; their internal computational dependencies are not re-audited.