Quotient-balanced subgroup property

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 metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

Definition

Suppose $a$ is a property of functions from a group to itself. Then, a normal subgroup $H$ of a group $G$ is said to satisfy the quotient-balanced subgroup property corresponding to $a$, if any map from $G$ to itself satisfying property $a$ sends every coset of $H$ to a coset of $H$, and the induced function from $G/H$ to itself also satisfies property $a$.