Group property-conditionally extensible automorphism

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This term is related to: extensible automorphisms problem
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Definition

Suppose $\alpha$ is a group property and $G$ is a group satisfying $\alpha$. An automorphism $\sigma$ of $G$ is termed extensible with respect to $\alpha$, or extensible conditional to $\alpha$, if for any group $H$ containing $G$ such that $H$ satisfies property $\alpha$, there is an automorphism $\sigma '$ of $H$ whose restriction to $G$ equals $\sigma$.

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-extensible automorphism for that subvariety.

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

Relation with other properties

Stronger properties

• Group property-conditionally pushforwardable automorphism

Related properties

• Group property-conditionally quotient-pullbackable automorphism

Other related notions

• Variety-extensible automorphism
• Own variety-extensible automorphism
• Extensibility operator
• Qualified extensibility operator