A finite modular lattice not embeddable in the lattice of formations of finite groups

Abstract

We give a negative answer to Kourovka Problem 13.51 by constructing a modular lattice with 184 elements that admits no injective map preserving binary meets and joins into the lattice of formations of finite groups. The counterexample is the dual of an explicit incidence lattice with 91 points and 91 lines. We prove a six-variable Desargues implication for formations by a finite argument using formation residuals and pullbacks of epimorphisms. The dual incidence lattice violates this implication. The obstruction includes the convention that the empty class is a formation and does not require preservation of the least or greatest element.