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Extensible local isomorphism

Definition

Suppose G is a group, A and B are subgroups of G, and \sigma:A \to B is an isomorphism of groups. We say that \sigma is an extensible local isomorphism if, for any group K containing G, there exists an automorphism \alpha of K such that the restriction of \alpha to A equals \sigma.

The extensible local isomorphisms conjecture states that an isomorphism of subgroups of G is an extensible local isomorphism if and only if it can be extended to an inner automorphism of G. This is a strong version of the extensible automorphisms conjecture.