The idea of a group is one of the great unifying ideas of mathematics. It arises in the study of symmetries, both of mathematical and of scientific objects. Very surprisingly, the examination of these symmetries leads to deep insights which are not available by direct inspection: while the notion of a group is very easy to explain, the applications of this concept do not at all lie on the surface. In mathematics the concept of a group is fundamental to the fields of differential geometry, topology, number theory and harmonic analysis, while in science this idea is essential in spectroscopy, crystalography, and atomic and particle physics. The importance of abstraction is nowhere more evident than in the concept of a group.
