Quotient-closed group property

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 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.
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View a list of other standard non-basic definitions

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 $p$ is said to be quotient-closed or Q-closed if whenever $G$ satisfies property $p$, and $N$ is a normal subgroup of $G$, the quotient group $G/N$ must also satisfy property $p$.

Relation with other metaproperties

Stronger metaproperties

• SQ-closed group property
• Quasivarietal group property
• Varietal group property