Abelian-quotient-pullbackable automorphism

Definition

Suppose $G$ is an abelian group and $\sigma$ is an automorphism of $G$. We say that $\sigma$ is an abelian-quotient-pullbackable automorphism of $G$ if, for any surjective homomorphism $\rho :K\to G$ from an abelian group $K$, there exists an automorphism $\sigma '$ of $K$ such that $\rho \circ \sigma '=\sigma \circ \rho$.

Relation with other properties

Related properties

• Abelian-quotient-pullbackable endomorphism
• Abelian-extensible automorphism
• Abelian-extensible endomorphism

Facts

• Free abelian group implies every automorphism is abelian-quotient-pullbackable
• Free abelian group implies every endomorphism is abelian-quotient-pullbackable
• Divisible abelian group implies every automorphism is abelian-extensible
• Divisible abelian group implies every endomorphism is abelian-extensible