# Epabelian group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A group is termed **epabelian** if it satisfies the following equivalent conditions:

- For any group with a central subgroup such that the quotient group is isomorphic to , must be an abelian group.
- The epicenter of equals .
- is an abelian group and its Schur multiplier (which is necessarily equal to its exterior square) is the trivial group.
- (
*Certainly necessary, not sure it is sufficient*): For any elements , either is cyclic or there exists a positive integer and an element such that and .

## 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.