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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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Contents 1 Statement 2 Definitions used 2.1 Extensible automorphism 2.2 Normal automorphism 3 Related facts 4 Facts used 5 Proof

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

• Inner implies extensible
• Extensible implies subgroup-conjugating
• Finite-extensible implies class-preserving
• Extensible automorphism-invariant equals normal

Facts used

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

Proof

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