Fully lattice-determined subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

Definition

A subgroup property $p$ is termed fully lattice-determined if, for any lattice-isomorphic groups $G_{1}, G_{2}$ , any lattice isomorphism $φ : L (G_{1}) \to L (G_{2})$ , and any subgroup $H_{1}$ of $G_{1}$ , $H_{1}$ satisfies $p$ in $G_{1}$ if and only if $φ (H_{1})$ satisfies $p$ in $G_{2}$ .