Normally finitely generated group

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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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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 Finitely generated group|FULL LIST, MORE INFO