Locally inner not implies inner
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., locally inner automorphism) need not satisfy the second automorphism property (i.e., inner automorphism)
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It is possible to have a group and an automorphism of such that is a locally inner automorphism of (i.e., the restriction of to every finitely generated subgroup of is inner on that subgroup) but is not an inner automorphism of .
The result follows from Fact (1). Note that the group for the statement we are trying to prove here will be the subgroup in the language on the page of Fact (1), and the automorphism will be an automorphism arising by restricting an inner automorphism of the ambient group that does not restrict to an inner automorphism.
Instead of using Fact (1) as a black box, we could also use either of the examples for it explicitly:
- The case that is the finitary symmetric group and is conjugation by an infinitary permutation.
- The case that is a restricted direct product of centerless groups and is conjugation by an element of the unrestricted direct product that has infinitely many non-identity coordinates.