= Countable torsion-free embedding with two conjugacy classes
Every nontrivial countable <torsion-free group> embeds in a countable torsion-free group with precisely two <conjugacy classes>, the identity and all nonidentity elements. At each stage, enumerate all ordered pairs of nonidentity elements of the current group and successively adjoin <HNN extensions> identifying their infinite cyclic subgroups. <Britton's lemma> preserves embeddings and <torsion in an HNN extension> preserves torsion-freeness. Repeat on the resulting countable union, so pairs involving newly introduced elements are eventually included. A final increasing union is countable and torsion-free, and any two nonidentity elements become conjugate at a later stage. No effective enumeration of nonidentity elements is claimed.
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