Group that is the normal closure of a singleton subset

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 defining ingredient::contranormal subgroup of $$G$$.
 * 3) There exists an element $$g \in G$$ such that the normal subgroup generated by $$\{ g \}$$ (or equivalently, the defining ingredient::normal closure of the subgroup it generates, $$\langle g \rangle$$) equals $$G$$.