Generation of finite groups from subgroups of coprime index

Abstract

Peer-review preprint and complete proof-audit trail for a claimed affirmative solution of Kourovka Notebook Problem 21.87. The manuscript proves, modulo the published inputs identified in the paper, that Lucchini's general generator bound sharpens from d+2 to d+1 for a finite group having a nonempty family of subgroups, each generated by at most d elements, whose indices have greatest common divisor 1. The proof reduces a minimal counterexample to a critical crown-based power and closes the bound by a Sylow-centralizer orbit count. The proof is deductive and uses no computer calculation. The cited results on critical crowns, conditional generation, and generation of finite simple groups depend on the classification of finite simple groups. This is a peer-review release, not an accepted or formally published article. Verified external finite-group-specialist review remains outstanding. A dated literature search found no earlier paper asserting the conclusion, but it does not establish novelty or priority.