# 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

### Collapse

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