Group property-conditionally quotient-pullbackable automorphism

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This term is related to: extensible automorphisms problem
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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