Uncountable set of 2-generated groups with exactly 2 conjugacy classes

You need to know: Group, identity element in a group, subgroup, isomorphic groups, conjugate elements in a group, conjugacy classes, countable and uncountable sets, notation for the number of elements in finite set S.

Background: A generating set of group G is a subset such that every can be written as a finite product of elements of S and their inverses. Group G is called finitely generated if it has finite generating set. In particular, it is called 2-generated if it has generating set S with . We say that group H can be embedded into group G, if G has a subgroup isomorphic to H. The order of non-identity element is the lowest positive integer n such that , with the convention that the order is infinite if no such n exists. For a group G, let be the set of all (finite) positive integers n such that there exists an element of order n.

The Theorem: On 2nd November 2004, Denis Osin submitted to arxiv a paper in which he proved, among other results, that any countable group H can be embedded into a 2-generated group G such that any two elements of the same order are conjugate in G and .

Short context: Applying the Theorem to any torsion-free group H (a group is called torsion-free if all non-identity elements in it have infinite order), we obtain that any countable torsion-free group can be embedded into a torsion-free 2-generated group with exactly 2 conjugacy classes. In turn, this implies that there exist uncountably many pairwise non-isomorphic torsion-free 2-generated groups with exactly 2 conjugacy classes. Before 2004, it was an open question whether there exists any finitely generated group G with which has exactly 2 conjugacy classes. The Theorem also implies that for any integer there are uncountably many pairwise nonisomorphic finitely generated groups with exactly n conjugacy classes.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.10 of this book for an accessible description of the Theorem.

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