Algebraic Characterization of Compact Abelian Groups

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The well-known Banach-Stone theorem gives a nice algebraic characterization of compact Hausdorff topological spaces. One might wonder if the topological spaces are also endowed with a (compatible) group structure whether one can obtain a similar characterization for them to be the same, i.e., to be topologically isomorphic. The purpose of this note is to give one such characterization which doesn't seem to have made its way into the literature yet. We begin by stating the result of Banach [1, p. 170] and Stone [7, p. 469] in a somewhat different form than the traditional version.