Epabelian group

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 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 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 $nb = 0$.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
cyclic group
locally cyclic group
periodic divisible abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group
epinilpotent group

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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.