Elliptic subgroup

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]

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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

Symbol-free definition

A subgroup of a group is termed elliptic if it forms an elliptic pair of subgroups with every subgroup of the group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed elliptic if for any subgroup $K$ of $G$, $(H,K)$ form an elliptic pair of subgroups. In other words, there exists an $n$ such that:

$⟨H,K⟩=(HK{)}^n:=HKHKHK\dots HK$

where each is written $n$ times.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
[[Weaker than::normal subgroup				|FULL LIST, MORE INFO
permutable subgroup	permutes with every subgroup			|FULL LIST, MORE INFO
subgroup of finite index				|FULL LIST, MORE INFO