# Every torsion-free group is a subgroup of a simple torsion-free group

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 (?): in other words, no non-identity element of has finite order. Then, there exists a Simple torsion-free group (?) containing .

## Facts used

## Proof

The proof follows directly from fact (1), and the observation that a group with two conjugacy classes is simple.