# Quotient-closed group property

From Groupprops

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

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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Quotient-closed group property, all facts related to Quotient-closed group property) |Survey articles about this | Survey articles about definitions built on this

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## Definition

### Symbol-free definition

A group property is said to be **quotient-closed** or **Q-closed** if any quotient of a group satisfying the property must also satisfy the property.

### Definition with symbols

A group property is said to be **quotient-closed** or **Q-closed** if whenever satisfies property , and is a normal subgroup of , the quotient group must also satisfy property .