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

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 $\sigma$ 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

Facts about injective endomorphisms

• Every injective endomorphism arises as the restriction of an inner automorphism

Applications

• 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 $⟨x⟩$ and $⟨y⟩$.
• 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