Natural Science, Mathematics, 2025
THE SET OF 2-GENERETED Câ -SIMPLE RELATIVELY FREE GROUPS HAS THE CARDINALITY OF THE CONTINUUM
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Submitted: 2025-01-29; Published: 2025-01-29
Š 2025 by author(s) and The Gufo Inc.
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(CC BY-NC 4.0).
Abstract
In this paper we prove that the set of non-isomorphic 2-generated Câ-simple relatively free groups has the cardinality of the continuum. A non-trivial identity is satisfied in any (not absolutely free) relatively free group. Hence, they cannot contain a non-abelian absolutely free subgroups. The question of the existence of Câ-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.