# Conditionally 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 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