# Cyclic implies abelian automorphism group

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., group whose automorphism group is abelian)

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## Statement

Any cyclic group is a group whose automorphism group is abelian: the automorphism group of a cyclic group is an abelian group.

## Related facts

## Proof

**Given**: A cyclic group with generator . Automorphisms of .

**To prove**: .

**Proof**: Suppose are two automorphisms of . Since is cyclic on , there exist integers such that . Thus, we have:

.

In particular, and are equal on the generator . Since generates , they must be equal as automorphisms.