We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski's theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions.

Infinite Abelian groups

Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

Slender modules over domains

In a paper published in this journal [1], J T Buckely, J C Lennox, B H Neumann and the authors considered the class of CF-groups, that G such that | H : Core G ( H )| is finite for all subgroups H  It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that | H : Core G ( H )| ≤ n for all subgroups H  The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite Whether locally graded CF-groups are BFC remains an open question In this direction, the following problems is posed Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF

Infinite Abelian groups

Citations

Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

Slender modules over domains

Locally graded groups with all subgroups normal-by-finite

Commutative rings with acc on n-generated ideals

On the lattice of cotorsion theories

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