# Residually cyclic 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 **residually cyclic** if it satisfies the following equivalent conditions:

- For every non-identity element, there is a normal subgroup of finite index of the whole group not containing that element, such that the quotient group is a cyclic group.
- The group is isomorphic to a subdirect product of cyclic groups.
- The group can be embedded as a subgroup in a direct product of cyclic groups.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Cyclic group | |FULL LIST, MORE INFO | |||

Finitely generated abelian group | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Residually finitely generated group | |FULL LIST, MORE INFO | |||

Abelian group | |FULL LIST, MORE INFO |