Quotient-transitive subgroup property

Definition with symbols
Let $$p$$ be a subgroup property. Then, $$p$$ is said to be quotient-transitive if whenever $$H$$ is a subgroup of $$G$$ such that there is a normal subgroup $$N$$ of $$G$$ contained in $$H$$ then the first of these implies the second:


 * $$N$$ satisfies $$p$$ in $$G$$ and $$H/N$$ satisfies $$p$$ in $$G/N$$.


 * $$H$$ satisfies $$p$$ in $$G$$.

In terms of the quotient-composition operator
A subgroup property $$p$$ is termed quotient-transitive if it is transitive with respect to the quotient-composition operator.

Remedies for lack of quotient-transitivity
There are three general ways to pass from a subgroup property to a quotient-transitive variation. These are analogous to the three ways to pass to an analogous transitive subgroup property.

Because of invariance
Any invariance property with respect to a quotient-hereditary function property is quotient-transitive. Examples are:


 * The property of being normal: This is the invariance property with respect to the quotient-hereditary function property of being an inner automorphism.
 * The property of being characteristic: This is the invariance property with respect to the quotient-hereditary function property of being an automorphism.
 * The property of being strictly characteristic: This is the invariance property with respect to the quotient-hereditary function property of being a surjective endomorphism.