Question:Inner automorphism conjugating element choice

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This question is about inner automorphism and conjugation. | See more questions about inner automorphism | See more questions about conjugation

Q: Given an inner automorphism, is there a unique element for which this is the conjugation operation? If the element is not unique, is there some natural way of picking an element?

A: The conjugating element need not be unique. In fact, the set of candidates for conjugating element for a given inner automorphism is a coset of the center. This is because the center is the kernel of the homomorphism G \to \operatorname{Aut}(G) given by conjugation action. The conjugating element is unique only in the case of a centerless group.

Moreover, there is no natural choice of conjugating element in general. In fact, even making a uniform choice may require use of the axiom of choice, since we need to choose infinitely many coset representatives.

Further, it is not necessary that we can make the choice in such a way that the product of two of the chosen representatives for the coset of center is again a chosen representative. In other words, we may not be able to choose our representatives so as to form a group. We can make such a choice only in a group whose center is a direct factor.