# Galois correspondence-closed 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

This article is about a general term. A list of important particular cases (instances) is available at Category:Galois correspondence-closed subgroup properties

## Contents

## Definition

A **Galois correspondence-closed subgroup property** is a subgroup property which arises in the following manner.

We start with a rule which, for every group, gives a binary relation between the group and another set constructed canonically from the group. The rule must be isomorphism-invariant, in the sense that any isomorphism of groups respects the binary relation.

The subgroup property we now get is the property of being a subgroup, which is *also* a closed subset of the group under the Galois correspondence induced by the binary relation.