# Quotient-transitive subgroup property

From Groupprops

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 metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Definition with symbols

Let be a subgroup property. Then, is said to be **quotient-transitive** if whenever is a subgroup of such that there is a normal subgroup of contained in then the first of these implies the second:

- satisfies in and satisfies in .

- satisfies in .

### In terms of the quotient-composition operator

A subgroup property is termed **quotient-transitive** if it is transitive with respect to the quotient-composition operator.

## Property theory

### Related metaproperties

### 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.