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
.