Group property-conditionally quotient-pullbackable automorphism

This term is related to: extensible automorphisms problem
View other terms related to extensible automorphisms problem | View facts related to extensible automorphisms problem

Definition

Suppose $\alpha$ is a group property and $G$ is a group satisfying $\alpha$. An automorphism $\sigma$ of $G$ is termed quotient-pullbackable with respect to $\alpha$, or quotient-pullbackable conditional to $\alpha$, if for any group $K$ and surjective homomorphism $\rho :K\to G$ such that $K$ satisfies property $\alpha$, there is an automorphism $\sigma '$ of $K$ such that $\rho \circ \sigma '=\sigma \circ \rho$.

For more information on the best known results and characterization, refer extensible automorphisms problem.

When the groups satisfying $\alpha$ form a subvariety of the variety of groups, this is equivalent to the notion of variety-quotient-pullbackable automorphism for that subvariety.

Also note that any inner automorphism is conditionally quotient-pullbackable with respect to any group property.

Relation with other properties

Related properties

• Group property-conditionally pushforwardable automorphism
• Group property-conditionally extensible automorphism