# 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]
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:

$\langle H,K \rangle = (HK)^n := HKHKHK \ldots HK$

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