# Normal subgroup-closed group property

From Groupprops

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This article defines a group metaproperty: a property that can be evaluated to true/false for any group property

View a complete list of group metaproperties

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

## Definition

A group property is termed **normal subgroup-closed** or **hereditary to normal subgroups** if it satisfies the following equivalent conditions:

- Whenever a group satisfies the property, so does every normal subgroup of it
- Whenever a group satisfies the property, so does every 2-subnormal subgroup of it
- Whenever a group satisfies the property, so does every subnormal subgroup of it