Ultrafilters and topologies on groups

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This book explores the interplay between ultrafilters and topologies on groups, demonstrating how ultrafilters help construct topologies with unique properties and how these topologies inform algebraic insights about ultrafilters. The content is divided into three parts. The first part (Chapters 1-5) introduces topological groups and ultrafilters without requiring the semigroup operation on ultrafilters. It includes constructions of significant topological groups, particularly an extremally disconnected group based on a Ramsey ultrafilter, and establishes that every infinite group has a nondiscrete zero-dimensional topology where all translations and inversions are continuous. The second part (Chapters 6-9) examines the Stone-Cêch compactification βG of a discrete group G, employing a technique involving local left groups and local homomorphisms. It proves that for a countable torsion-free group G, βG lacks nontrivial finite groups and investigates its ideal structure, revealing that every infinite Abelian group G has 22|G| minimal right ideals in βG. The final part utilizes the semigroup βG to construct almost maximal topological and left topological groups, analyzing their ultrafilter semigroups and characterizing projectives in finite semigroups. It shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω-resolvable, allowing partitioning into ω subsets. The book concludes with open pro