Subpronormal subgroup

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

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Subpronormal subgroup, all facts related to Subpronormal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties

Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed subpronormal if there exists an ascending chain:

$H=H_{0}\leq H_{1}\leq \ldots \leq H_{n}=G$

such that each $H_{i}$ is a pronormal subgroup of $H_{i+1}$.

Formalisms

In terms of the subordination operator

This property is obtained by applying the subordination operator to the property: pronormal subgroup
View other properties obtained by applying the subordination operator

Relation with other properties

Stronger properties

• Pronormal subgroup
• Subnormal subgroup
• Subabnormal subgroup
• Submaximal subgroup
• Subgroup of finite index

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity