Join of finitely many pronormal subgroups

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

Definition

A subgroup $H$ of a group $G$ is termed a join of finitely many pronormal subgroups if there exist subgroups $H_{1},H_{2},\dots ,H_{n}$ of $G$ (with $n$ a natural number) such that $H$ is the join of all the $H_{i}$s.

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
pronormal subgroup	any conjugate subgroup is conjugate to it in their join	(by definition)	pronormality is not finite-join-closed	|FULL LIST, MORE INFO
normal subgroup	equals all its conjugate subgroups	(via pronormal)		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
join of pronormal subgroups	join of (possibly infinitely many) pronormal subgroups	(by definition)	?	|FULL LIST, MORE INFO