Subpronormal subgroup

This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
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Definition

Definition with symbols

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

$H=H_0\le H_1\le \dots \le H_n=G$

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