Epabelian group

Definition
A group $$G$$ is termed epabelian if it satisfies the following equivalent conditions:


 * 1) For any group $$K$$ with a central subgroup $$H$$ such that the quotient group $$K/H$$ is isomorphic to $$G$$, $$K$$ must be an abelian group.
 * 2) The defining ingredient::epicenter of $$G$$ equals $$G$$.
 * 3) $$G$$ is an abelian group and its Schur multiplier (which is necessarily equal to its exterior square) is the trivial group.
 * 4) (Certainly necessary, not sure it is sufficient): For any elements $$a,b \in G$$, either $$\langle a,b \rangle$$ is cyclic or there exists a positive integer $$n$$ and an element $$c \in G$$ such that $$nc = a$$ and $$nb = 0$$.

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Any finitely generated epabelian group is cyclic. This follows from the more general fact that finitely generated and epinilpotent implies cyclic.