Extensible implies normal

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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., extensible automorphism) must also satisfy the second automorphism property (i.e., normal automorphism)
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Get more facts about extensible automorphism|Get more facts about normal automorphism

Statement

Every extensible automorphism of a group is a normal automorphism: it sends every normal subgroup of the group to itself.

Definitions used

Extensible automorphism

Further information: Extensible automorphism

Normal automorphism

Further information: Normal automorphism

Related facts

Facts used

  1. Extensible implies subgroup-conjugating
  2. Subgroup-conjugating implies normal

Proof

The proof follows directly by piecing together facts (1) and (2).