# Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes

From Groupprops

This article gives the statement, and possibly proof, of an embeddability theorem: a result that states that any group of a certain kind can be embedded in a group of a more restricted kind.

View a complete list of embeddability theorems

## Statement

Suppose is a torsion-free group. Then, there exists a torsion-free group with two conjugacy classes that contains . Further, if is countable, we can choose to be countable.

## Related facts

### Generalizations

## Facts used

- Isomorphic iff potentially conjugate: Suppose is a collection of isomorphisms between subgroups of . Then, there exists a group containing , with elements , such that conjugation by , restricted to , is equal to the isomorphism . Moreover, if is torsion-free, we can choose to be torsion-free.

## Proof

**Given**: A torsion-free group .

**To prove**: There exists a torsion-free group containing such that has two conjugacy classes.

**Proof**:

- There exists a torsion-free group containing such that any two non-identity elements of are conjugate in : Let be the set of all pairs of non-identity elements in , and be the isomorphism between the cyclic subgroups generated by the first element of the pair and the cyclic subgroup generated by the second element of the pair (sending the generator to the generator). Note that such an isomorphism exists because all non-identity elements have the same order (infinite). By fact (1), there exists a group satisfying the required conditions.
- There exists a chain where any two non-identity elements of are conjugate in : This follows by repeating the previous step.
- Define as the ascending union (or direct limit) of the s. Then, any two non-identity elements of are conjugate in .