Quotient-pullbackable equals inner

This article gives a proof/explanation of the equivalence of multiple definitions for the term inner automorphism
View a complete list of pages giving proofs of equivalence of definitions

This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problem | View terms related to Extensible automorphisms problem

Statement

The following are equivalent for an automorphism $\sigma$ of a group $G$:

1. The automorphism is a quotient-pullbackable automorphism: For any homomorphism $\rho :H\to G$, there is an automorphism $\varphi$ of $H$, $\rho \circ \varphi =\sigma \circ \rho$.
2. The automorphism is an inner automorphism.

Definitions used

Quotient-pullbackable automorphism

An automorphism $\sigma$ of a group $G$ is termed quotient-pullbackable if given any surjective homomorphism $\rho :H\to G$ there is an automorphism $\varphi$ of $H$ such that $\rho \circ \varphi =\sigma \circ \rho$.

Inner automorphism

Further information: Inner automorphism

An automorphism $\sigma$ of a group $G$ is termed an inner automorphism if there exists $g\in G$ such that $\sigma =c_{g}=x\mapsto gxg^{-1}$.

Related facts

• Extensible implies inner
• Finite-extensible implies inner
• Finite-quotient-pullbackable implies inner

References

• 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}