# Isomorphic iff potentially conjugate

## Statement

### 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

### Facts about automorphisms extending to inner automorphisms

• Inner automorphism to automorphism is right tight for normality: In other words, if  is an automorphism of , there exists a group  containing  as a normal subgroup, and an inner automorphism of  whose restriction to  equals .
• Left transiter of normal is characteristic: A direct application of the fact that any automorphism of a group extends to an inner automorphism in a bigger group containing it as a normal subgroup. This says that  is such that (whenever  is normal in ,  is also normal in ) if and only if  is characteristic in .
• Characteristic of normal implies normal