# Isomorphic iff potentially conjugate

## Contents

## 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

### For finite groups

- Isomorphic iff potentially conjugate in finite: This construction works when the original group is finite and yields a bigger group that is also finite.

### 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

### Facts about injective endomorphisms

### Applications

- Same order iff potentially conjugate: are such that have the same order if and only if then there is a group containing in which and are conjugate elements. This is a direct application based on looking at the cyclic subgroups and .
- Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes
- Every torsion-free group is a subgroup of a simple torsion-free group