Isomorphic iff potentially conjugate

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Statement

For just one pair of isomorphic subgroups

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

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


For a collection of many pairs of isomorphisms between subgroups

Suppose G is a group, I is an indexing set, and H_i \cong K_i are pairs of isomorphic subgroups of G for each i \in I. Let \sigma_i: H_i \to K_i be an isomorphism for each i \in I.

Then, there exists a group L containing G as a subgroup such that H_i and K_i are conjugate subgroups in L for each i \in I. More specifically, we can find g_i, i \in I such that the map induced by conjugation by g_i induces the isomorphism \sigma_i.

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

Related facts

For finite groups

Facts about automorphisms extending to inner automorphisms

Facts about injective endomorphisms

Applications