Conditionally lattice-determined subgroup property

Definition

A subgroup property $p$ is termed conditionally lattice-determined if, for any group $G$, any lattice automorphism $\varphi: L(G) \to L(G)$ of the lattice of subgroups $L(G)$, and any subgroup $H$ of $G$, $H$ satisfies $p$ if and only if $\varphi(H)$ satisfies $p$.

The use of the qualifier conditionally is to contract with fully lattice-determined subgroup property, where we allow the ambient group to also vary.

Relation with other metaproperties

Stronger metaproperties

Metaproperty Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully lattice-determined subgroup property determined up to lattice isomorphism (between possibly non-isomorphic ambient groups) |FULL LIST, MORE INFO