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Let      S S       be the structure semigroup of a measure algebra      M ( G ) M(G)       and      K K       be the union of all maximal groups of      S S      . Taylor proposed the following problem: Are there L. C. A. groups      G G       with nontrivial measures concentrated on      K K      ? The purpose of this paper is to give a positive solution to this problem.

On a problem of J. L. Taylor