Power automorphism not implies universal power automorphism
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., power automorphism) need not satisfy the second automorphism property (i.e., universal power automorphism)
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Let be a group. An automorphism of is termed a power automorphism of if, for every , there exists a such that .
Universal power automorphism
Let be a group. An automorphism of is termed a universal power automorphism (or a uniform power automorphism) if there exists a such that for all .
Example of the quaternion group
In the quaternion group:
conjugation by is a power automorphism that is not a universal power automorphism.
- It is a power automorphism: It fixes , and sends the other elements to their inverses.
- It is not a universal power automorphism: It sends the elements to their inverses. Since each of these elements have order four, any for which this automorphism is the power map, must be congruent to mod . On the other hand, since it fixes , which also has order four, must be modulo , a contradiction.