Inner implies infinity-extensible

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This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., inner automorphism) must also satisfy the second automorphism property (i.e., infinity-extensible automorphism)
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Statement

Any inner automorphism of a group is an infinity-extensible automorphism.

Facts used

  1. Inner is extensibility-stable: Any inner automorphism of a subgroup can be extended to an inner automorphism of the whole group.

Proof

The proof follows from fact (1), and the fact that inner automorphisms are automorphisms.