Cyclic implies abelian automorphism group
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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.