# Cyclic implies abelian

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., cyclic group) must also satisfy the second group property (i.e., abelian group)

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## Contents

## Statement

Any cyclic group (i.e., any group generated by only one element) is an abelian group (i.e., any two elements in it commute).

## Related facts

### Using weaker hypotheses to deduce abelianness

- Locally cyclic implies abelian
- Residually cyclic implies abelian
- Generating set in which any two elements commute implies abelian
- Cyclic over central implies abelian