Finite-extensible implies inner
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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Contents
Statement
Suppose is a finite group and is a finite-extensible automorphism of : in other words, extends to an automorphism of for any finite group containing . Then, is an inner automorphism of .
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
Facts used
- Every finite group is the Fitting quotient of a p-dominated group for any prime p not dividing its order: Suppose is a finite group and is a prime not dividing the order of . Then, there exists a p-dominated group with as Fitting quotient: in other words, there exists a finite complete group such that the Fitting subgroup is a -group, and is a subgroup of such that .
Proof
Given: A finite group , a finite-extensible automorphism of .
To prove: is inner.
Proof: Let be a prime not dividing the order of . Consider the group constructed by fact (1). Since is finite-extensible, extends to an automorphism of . Further, since is complete, there exists such that is conjugation by .
Let be the retraction with kernel . Note that conjugation by preserves , hence it induces a conjugation map on as a quotient, namely, conjugation by the element . However, since the restriction of to the subgroup is the identity map, we conclude that conjugation by has the same effect on as conjugation by . In particular, equals conjugation by , 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 link}^{More 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 copy}^{More info}