# Conditionally lattice-determined subgroup property

From Groupprops

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 metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Definition

A subgroup property is termed **conditionally lattice-determined** if, for any group , any lattice automorphism of the lattice of subgroups , and any subgroup of , satisfies if and only if satisfies .

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 |