Interval enforceable properties of finite groups

TL;DR

This paper classifies certain group properties based on whether they can be inferred from the subgroup lattice structure, introducing notions of interval enforceability and core-free interval enforceability.

Contribution

It introduces a new classification framework for group properties based on subgroup lattice intervals and explores their implications for open problems in universal algebra.

Findings

Some group properties are interval enforceable

Other properties are weakly core-free interval enforceable

Potential resolution of an open problem in universal algebra

Abstract

We propose a classification of group properties according to whether they can be deduced from the assumption that a group's subgroup lattice contains an interval isomorphic to some lattice. We are able to classify a few group properties as being "interval enforceable" in this sense, and we establish that other properties satisfy a weaker notion of "core-free interval enforceable." We also show that if there exists a group property and its negation that are both core-free interval enforceable, this would settle an important open question in universal algebra.