P-quotient-pullbackable implies inner

Statement

Suppose $p$ is a prime number and $P$ is a p-group, i.e., a group in which the order of every element is a power of $p$ and $\sigma$ is an automorphism of $P$.

Suppose that, for any p-group $Q$ and surjective homomorphism $f:Q\to P$, there exists an automorphism ${\sigma }^{\prime }$ of $Q$ such that $f\circ {\sigma }^{\prime }=\sigma \circ f$. Then, $\sigma$ is an inner automorphism of $P$.

References

Textbook 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}
• 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}