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The aim of this paper is to present some new and essential facts about group  generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to . Additionally, we show that there exists a family  G  (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than  and is composed only from the permutations preserving the sum of series. This result substantially strengthens some old Pleasants’ result.

Some new facts about group 𝒢 generated by the family of convergent permutations