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

1. 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.
2. The group is isomorphic to a subdirect product of cyclic groups.
3. 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