Group that is the characteristic closure of a singleton subset

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 defining ingredient::characteristic subgroups.
 * 2) There exists an element $$g \in G$$ such that the defining ingredient::characteristic closure of $$\{ g \}$$ (or equivalently, the characteristic closure of the cyclic subgroup $$\langle g \rangle$$) in $$G$$ is the whole group $$G$$.