Exceptional isomorphism
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members "a" and "b" of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such isomorphisms. These coincidences are at times considered a matter of trivia, but in other respects they can give rise to other phenomena, notably exceptional objects. In the following, coincidences are listed wherever they occur. Groups. Finite simple groups. The exceptional isomorphisms between the series of finite simple groups mostly involve projective special linear groups and alternating groups, and are: Alternating groups and  symmetric groups. There are coincidences between symmetric/alternating groups and small groups of Lie type/polyhedral groups: These can all be explained in a systematic way by using linear algebra (and the action of formula_16 on affine formula_17-space) to define the isomorphism going from the right side to the left side. (The above isomorphisms for formula_18 and formula_19 are linked via the exceptional isomorphism formula_20.) There are also some coincidences with symmetries of regular polyhedra: the alternating group A agrees with the icosahedral group (itself an exceptional object), and the double cover of the alternating group A is the  binary icosahedral group. Trivial group. The trivial group arises in numerous ways. The trivial group is often omitted from the beginning of a classical family. For instance: Spheres. The spheres "S", "S", and "S" admit group structures, which can be described in many ways: Spin groups. In addition to formula_29, formula_30 and formula_31 above, there are isomorphisms for higher dimensional spin groups: Also, Spin(8) has an exceptional order 3 triality automorphism Coxeter–Dynkin diagrams. There are some exceptional isomorphisms of Dynkin diagrams, yielding isomorphisms of the corresponding Coxeter groups and of polytopes realizing the symmetries, as well as isomorphisms of  lie algebras whose root systems are described by the same diagrams. These are: