Frankl's Conjecture for subgroup lattices

TL;DR

This paper proves that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture, extending the result to lattices with a modular coatom, including supersolvable and dually semimodular lattices.

Contribution

It establishes the validity of Frankl's Conjecture for subgroup lattices of finite groups and for a broader class of lattices with a modular coatom, introducing a new technical result.

Findings

Subgroup lattices of finite groups satisfy Frankl's Conjecture.

Lattices with a modular coatom, including supersolvable and dually semimodular lattices, also satisfy the conjecture.

A new technical result applicable to these classes is presented.

Abstract

We show that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.