# Normally finitely generated group

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

QUICK PHRASES: normal closure in itself of finite subset, has finitely generated contranormal subgroup

### Equivalent definitions in tabular format

A group is termed a **normally finitely generated group** or **finitely normally generated group** or **normal closure in itself of finite subset** if it satisfies the following equivalent conditions:

No. | Shorthand | A group is normally finitely generated if... | A group is normally finitely generated if ... |
---|---|---|---|

1 | normally generated by finite subset | there is a finite subset such that the group equals the normal subgroup generated by it | there is a finite subset of such that . |

2 | finitely generated contranormal subgroup | it has a contranormal subgroup (i.e., normal closure the whole group) that is a finitely generated group | there is a subgroup of such that and is finitely generated. |

3 | normal closure in itself of finite subset | it is a normal closure of finite subset in itself | (same as 1) |

## Relation with other properties

### Stronger properties

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

Simple group | no proper nontrivial normal subgroup | |FULL LIST, MORE INFO | ||

Group that is the normal closure of a singleton subset | has a cyclic contranormal subgroup | |FULL LIST, MORE INFO | ||

Finitely generated group | |FULL LIST, MORE INFO | |||

Finite group | Finitely generated group|FULL LIST, MORE INFO |