# Normal subgroup of finite index

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup of finite index

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

A **normal subgroup of finite index** in a group is a subgroup satisfying the following equivalent conditions:

- It is normal and its index in the whole group is finite
- It is the kernel of a homomorphism to a finite group
- It is the normal core of a subgroup of finite index

### Equivalence of definitions

The equivalence of definitions (1) and (2) follows from the first isomorphism theorem. The equivalence with definition (3) follows from Poincare's theorem.

## Relation with other properties

### Stronger properties

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

characteristic subgroup of finite index | characteristic subgroup and its index in the whole group is finite. | characteristic implies normal | any finite example for normal not implies characteristic | |FULL LIST, MORE INFO |

normal subgroup of finite group | normal subgroup and the whole group is a finite group. | normal subgroup of finite index|normal subgroup of finite group}} |

### Weaker properties

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

subgroup of finite index | Subnormal subgroup of finite index|FULL LIST, MORE INFO | |||

normal subgroup with finitely generated quotient group | |FULL LIST, MORE INFO | |||

quotient-powering-invariant subgroup | normal subgroup of finite index implies quotient-powering-invariant | take any finite normal subgroup in an infinite group; see finite normal implies quotient-powering-invariant | |FULL LIST, MORE INFO | |

powering-invariant subgroup | Intermediately powering-invariant subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO |