Isomorphic iff potentially conjugate

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For just one pair of isomorphic subgroups

Suppose is a group and are isomorphic groups, i.e., there is an isomorphism of groups, say , from to (Note that this isomorphism need not arise from an automorphism of , so and need not be automorphic subgroups).

Then, there exists a group containing such that are conjugate subgroups inside , and the induced isomorphism from to by that conjugating element equals .

For a collection of many pairs of isomorphisms between subgroups

Suppose is a group, is an indexing set, and are pairs of isomorphic subgroups of for each . Let be an isomorphism for each .

Then, there exists a group containing as a subgroup such that and are conjugate subgroups in for each . More specifically, we can find such that the map induced by conjugation by induces the isomorphism .

Moreover, there is a natural construction of such a group , called a HNN-extension. In the case that is an torsion-free group, we can ensure that the group is also torsion-free.

Related facts

For finite groups

Facts about automorphisms extending to inner automorphisms

Facts about injective endomorphisms