  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 prescribe inner automorphism group. The latter
theorem suggests a new approach to the classification of finite connected
quandles.
