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

Contents 1 Statement 2 Related facts 2.1 Weaker facts 3 Facts used 4 Proof 5 References 5.1 Journal references

Statement

Suppose H is a finite group and σ is a finite-extensible automorphism of H: in other words, σ extends to an automorphism of G for any finite group G containing H. Then, σ is an inner automorphism of H.

Note that since any inner automorphism is extensible, this says that the property of being finite-extensible is equivalent to the property of being inner for a finite group.

Related facts

• Extensible implies inner
• Finite-quotient-pullbackable implies inner
• Hall-semidirectly extensible implies inner
• Finite solvable-extensible implies inner

Weaker facts

• Finite-extensible implies class-preserving
• Finite-extensible implies subgroup-conjugating

Facts used

1. Every finite group is the Fitting quotient of a p-dominated group for any prime p not dividing its order: Suppose H is a finite group and p is a prime not dividing the order of H. Then, there exists a p-dominated group G with H as Fitting quotient: in other words, there exists a finite complete group G such that the Fitting subgroup F(G) is a p-group, and H is a subgroup of G such that .

Proof

Given: A finite group H, a finite-extensible automorphism σ of H.

To prove: σ is inner.

Proof: Let p be a prime not dividing the order of H. Consider the group G constructed by fact (1). Since σ is finite-extensible, σ extends to an automorphism σ' of G. Further, since G is complete, there exists such that σ' is conjugation by g.

Let be the retraction with kernel F(G). Note that conjugation by g preserves F(G), hence it induces a conjugation map on H as a quotient, namely, conjugation by the element . However, since the restriction of ρ to the subgroup H is the identity map, we conclude that conjugation by g has the same effect on H as conjugation by ρ(g). In particular, σ equals conjugation by ρ(g), and hence is inner.

References

Journal references

• On inner automorphisms of finite groups by Martin R. Pettet, Proceedings of the American Mathematical Society, Volume 106,Number 1, Page 87 - 90(May 1989): ^{JSTOR linkMore info}
• Characterizing inner automorphisms of groups by Martin R. Pettet, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 55,Number 5, Page 422 - 428(Year 1990): ^{Springerlink official copyMore info}