# Normally finitely generated group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## 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 $G$ 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 $A$ of $G$ such that $\langle A^G \rangle = G$.
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 $H$ of $G$ such that $H^G = G$ and $H$ 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