κ-fold transitive groups

Abstract

An abelian group G of type 0 is called κ-fold transitive for some cardinal κ > 0 if for any pair of pure elements x, y ∈ G there exist exactly κ-many ϕ ∈ Aut G such that xϕ = y. We show the existence of large κ-fold transitive groups for every κ ≥ ℵ 0 assuming V=L and ZFC respectively. 2000 Mathematics Subject Classification: 13C05, 13C10, 13C13, 20K15, 20K25, 20K30; 03E05, 03E35.