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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Statement

Suppose $H$ is a finite group and $\sigma$ is a finite-extensible automorphism of $H$: in other words, $\sigma$ extends to an automorphism of $G$ for any finite group $G$ containing $H$. Then, $\sigma$ 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 $G=F(G)⋊H$.

Proof

Given: A finite group $H$, a finite-extensible automorphism $\sigma$ of $H$.

To prove: $\sigma$ is inner.

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

Let $\rho :G\to H$ 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 $\rho (g)\in H$. However, since the restriction of $\rho$ to the subgroup $H$ is the identity map, we conclude that conjugation by $g$ has the same effect on $H$ as conjugation by $\rho (g)$. In particular, $\sigma$ equals conjugation by $\rho (g)$, 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}