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
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				|FULL LIST, MORE INFO