 One of the most important roles played by symmetry is that of classification — for example, the classification of crystals using their remarkable and varied symmetry properties. In contemporary physics, the best example of this role of symmetry is the classification of elementary particles by means of the irreducible representations of the fundamental physical symmetry groups, a result first obtained by Wigner in his famous paper of 1939 on the unitary representations of the inhomogeneous Lorentz group. When a symmetry classification includes all the necessary properties for characterizing a given type of physical object (for example, all necessary quantum numbers for characterizing a given type of particle), we have the possibility of defining types of entities on the basis of their transformation properties. This has led philosophers of science to explore a structuralist approach to the entities of modern physics, in particular a group-theoretical account of objects (see the contributions in Castellani, 1998, Part II). The relation between symmetry, group and structure and its exploitation in structuralist approaches, especially in the framework of structural realism, is a much debated issue. See, for example, Roberts (2011), French (2014) and Caulton (2015), and the SEP entry  Structural Realism.  