P-quotient-pullbackable automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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Definition

Let $P$ be a group of prime power order, i.e., a finite $p$-group for some prime $p$. An automorphism $\sigma$ of $P$ is termed a p-quotient-pullbackable automorphism if, given any finite $p$-group $Q$ and a surjective homomorphism $\rho :Q\to P$, there exists an automorphism $\sigma '$ of $Q$ that is a pullback of $\sigma$; in other words, $\rho \circ \sigma '=\sigma \circ \rho$.

Relation with other properties

Stronger properties

• Inner automorphism of a finite $p$-group.

Weaker properties

• Cofactorial automorphism: Any $p$-quotient-pullbackable automorphism of a finite $p$-group is itself a $p$-automorphism. For full proof, refer: p-quotient-pullbackable automorphism implies p-automorphism

Facts

• p-quotient-pullbackable automorphism of elementary abelian group is trivial: This is an immediate corollary of Bryant-Kovacs theorem.