Normal-extensible not implies extensible

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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., normal-extensible automorphism) need not satisfy the second automorphism property (i.e., extensible automorphism)
View a complete list of automorphism property non-implications | View a complete list of automorphism property implications
Get more facts about normal-extensible automorphism|Get more facts about extensible automorphism

Statement

There exists a group G and a normal-extensible automorphism \sigma of G such that \sigma is not an extensible automorphism.

Facts used

  1. Finite-extensible implies class-preserving
  2. Centerless and maximal in automorphism group implies every automorphism is normal-extensible

Proof

Let G be the alternating group:A4: in other words, G is the alternating group on the set \{ 1,2,3,4 \}. The automorphism group K of G can be identified naturally with the symmetric group on \{ 1,2,3, 4 \}, with G embedded in it as inner automorphisms. By fact (2), every automorphism of G (including the outer automorphisms) is normal-extensible.

On the other hand, none of the outer automorphisms of G preserves conjugacy classes in G. Thus, by fact (1), none of the outer automorphisms of G is finite-extensible. In other words, for any outer automorphism \sigma of G, there exists a finite group L containing G such that \sigma does not extend to L. In particular, \sigma is not extensible, and we are done.