In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class.
Some facts: For infinite groups, an example of a class automorphism that is not inner is the following: take the finitary symmetric group on countably many elements and consider conjugation by an infinitary permutation.
This conjugation defines an outer automorphism on the group of finitary permutations.
Finding a class automorphism in the stability group that is not inner boils down to finding a cocycle for the action that is locally a coboundary but is not a global coboundary.
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