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