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    This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU "Brainstorming and Barnstorming". The main results are a decomposition theorem for quandles in terms of an operation of "semidisjoint union" showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribed inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.

    AMSC: 57M27, 20N99