Quotient-transitive subgroup property

From Groupprops
Jump to: navigation, search
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 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

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: