# Group having no proper subgroup of finite index

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **group having no proper subgroup of finite index** is a group satisfying the following equivalent conditions:

- It has no proper subgroup of finite index.
- It has no proper normal subgroup of finite index.

Apart from the trivial group, any group with this property must be infinite.

### Equivalence of definitions

The equivalence follows from Poincare's theorem.

## Relation with other properties

### Stronger properties

- Infinite simple group

### Weaker properties

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

group with finitely many homomorphisms to any finite group | ||||

group in which every subgroup of finite index has finitely many automorphic subgroups | ||||

group in which every subgroup of finite index is normal |