Our work begins with the development of a topological framework for the key elements of our subject. The first section introduces the category of topological groups and their fundamental properties. We treat, in particular, uniform continuity, separation properties, and quotient spaces. In the second section we narrow our focus to locally compact groups, which serve as the locale for the most important mathematical phenomena treated subsequently. We establish the essential deep feature of such groups: the existence and uniqueness of Haar measure; this is fundamental to the development of abstract harmonic analysis. The last two sections further specialize to profinite groups, giving a topological characterization, a structure theorem, and a set of results roughly analogous to the Sylow Theorems for finite groups. The prerequisites for this discussion will be found in almost any first-year graduate courses in algebra and analysis.
KeywordsNormal Subgroup Open Neighborhood Topological Group Haar Measure Closed Subgroup
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