Group that is the normal 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 a group that is the normal closure of a singleton subset if it satisfies the following equivalent conditions:

1. $G$ is not the union of all its proper normal subgroups.
2. There exists a cyclic contranormal subgroup of $G$.
3. There exists an element $g\in G$ such that the normal subgroup generated by $\{g\}$ (or equivalently, the normal closure of the subgroup it generates, $\langle g\rangle$) equals $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Simple group
Cyclic group
Group with two conjugacy classes
Group having a cyclic conjugate-dense subgroup

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group that is the characteristic closure of a singleton subset
Group having an abelian contranormal subgroup
Normally finitely generated group