Normal-extensible not implies extensible
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)
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- Finite-extensible implies class-preserving
- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
Let be the alternating group:A4: in other words, is the alternating group on the set . The automorphism group of can be identified naturally with the symmetric group on , with embedded in it as inner automorphisms. By fact (2), every automorphism of (including the outer automorphisms) is normal-extensible.
On the other hand, none of the outer automorphisms of preserves conjugacy classes in . Thus, by fact (1), none of the outer automorphisms of is finite-extensible. In other words, for any outer automorphism of , there exists a finite group containing such that does not extend to . In particular, is not extensible, and we are done.