Quotient-transitive subgroup property

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

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.

Property theory

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.

Naturally arising quotient-transitive subgroup properties

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.