Quotient-pullbackable equals 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., quotient-pullbackable automorphism) must also satisfy the second automorphism property (i.e., inner automorphism)
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Statement

Any quotient-pullbackable automorphism of a group is an inner automorphism.

Definitions used

Quotient-pullbackable automorphism

Further information: 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 $\phi$ of $H$ such that $\rho \circ \phi =\sigma \circ \rho$.

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 $\sigma =c_g=x\mapsto gx{g}^{-1}$.

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}