Group that is the characteristic closure of a singleton subset

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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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Definition

A group $G$ is termed the characteristic closure of a singleton subset if it satisfies the following equivalent conditions:

1. $G$ is not the union of all its proper characteristic subgroups.
2. There exists an element $g\in G$ such that the characteristic closure of $\{g\}$ (or equivalently, the characteristic closure of the cyclic subgroup $\langle g\rangle$) in $G$ is the whole group $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Cyclic group	generated by one element			|FULL LIST, MORE INFO
Group that is the normal closure of a singleton subset	normal closure of a singleton subset			|FULL LIST, MORE INFO
Characteristically simple group	no proper nontrivial characteristic subgroups			|FULL LIST, MORE INFO
Simple group	no proper nontrivial normal subgroups			|FULL LIST, MORE INFO
Finite abelian group	finite and abelian			|FULL LIST, MORE INFO
Free group				|FULL LIST, MORE INFO
Reduced free group				|FULL LIST, MORE INFO