Generation of finite groups from subgroups of coprime index

Version 1.2.1 of the preprint and supporting source records for Kourovka Notebook Problem 21.87. If a finite group G has a nonempty family of subgroups, each generated by at most d elements, whose indices have greatest common divisor one, then G can be generated by d+1 elements. The proof reduces a least counterexample to a critical crown-based power with nonabelian socle. Conditional generation gives a lower bound on its multiplicity, while a subgroup containing a Sylow 2-subgroup gives an incompatible upper bound through pointwise automorphism centralizers and coordinate-tuple orbits. This six-page revision compresses standard arguments and places the external theorem citations at their points of use. It compares the proof with DeCaro's same-theorem preprint and identifies the distinct pointwise Sylow-centralizer argument. The mathematical statements and bounds are unchanged.