# Isomorphic iff potentially conjugate

## 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

### Facts about automorphisms extending to inner automorphisms

• Inner automorphism to automorphism is right tight for normality: In other words, if $\sigma$ is an automorphism of $G$, there exists a group $L$ containing $G$ as a normal subgroup, and an inner automorphism of $L$ whose restriction to $G$ equals $\sigma$.
• 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 $H \le G$ is such that (whenever $G$ is normal in $L$, $H$ is also normal in $L$) if and only if $H$ is characteristic in $G$.
• Characteristic of normal implies normal

• Same order iff potentially conjugate: $x,y \in G$ are such that $x,y$ have the same order if and only if then there is a group $L$ containing $G$ in which $x$ and $y$ are conjugate elements. This is a direct application based on looking at the cyclic subgroups $\langle x \rangle$ and $\langle y \rangle$.