# Elliptic subgroup

From Groupprops

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

## Contents

## 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 of a group is termed **elliptic** if for any subgroup of , form an elliptic pair of subgroups. In other words, there exists an such that:

where each is written times.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup | Permutable subgroup|FULL LIST, MORE INFO | |||

permutable subgroup | permutes with every subgroup | |FULL LIST, MORE INFO | ||

subgroup of finite index | Subgroup of finite double coset index|FULL LIST, MORE INFO |