# Automorphism-based relation-implication-expressible 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

## Contents

## Statement

Suppose and are two group-closed properties of automorphisms from a group to itself. Let denote the following relation between subgroups: two subgroups satisfy if there is an automorphism of satisfying property and sending to . Analogously, define . The automorphism-based relation-implication-expressible subgroup property for and is the property with relation implication expression:

.

In other words, satisfies property in if whenever satisfy (i.e., there is an automorphism with property sending to ), also satisfy property (i.e., there is an automorphism with property sending to ).

## Examples

Here are some examples:

- Characteristic subgroup: Here, the two properties are automorphism and identity map.
- Normal subgroup: Here, the two properties are inner automorphism and identity map.
- Automorph-conjugate subgroup: Here, the two properties are automorphism and inner automorphism.