Group property-conditionally pushforwardable automorphism

Definition

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

Also note that any inner automorphism is conditionally pushforwardable with respect to any group property.

Relation with other properties

Weaker properties

• Group property-conditionally extensible automorphism

Related properties

• Group property-conditionally quotient-pullbackable automorphism