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

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

Facts used

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

Proof

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