Universal power not implies class-preserving
From Groupprops
This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., universal power automorphism) need not satisfy the second automorphism property (i.e., class-preserving)
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Contents
Statement
A universal power automorphism of a group (i.e., an automorphism obtained by taking the power for some integer
) need not be a class-preserving automorphism: it need not preserve conjugacy classes of elements.
Related facts
Stronger facts
Corollaries
Proof
Example of an Abelian group
Consider the cyclic group of order , where
is an odd prime. The inverse map is a universal power automorphism of this group, and is not the identity map. Since the group is Abelian, the conjugacy classes are singleton, so we have a universal power automorphism that is not class-preserving.
The example generalizes to any Abelian group whose exponent is bigger than two.