Exceptional isomorphism
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj 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:- the smallest non-abelian simple group – icosahedral symmetry;
- the second-smallest non-abelian simple group – PSL;
- between a projective special orthogonal group and a projective symplectic group.
Alternating groups and symmetric groups
These can all be explained in a systematic way by using linear algebra
to define the isomorphism going from the right side to the left side.
There are also some coincidences with symmetries of regular polyhedra: the alternating group A5 agrees with the icosahedral group, and the double cover of the alternating group A5 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:- , the cyclic group of order 1;
- , the alternating group on 0, 1, or 2 letters;
- , the symmetric group on 0 or 1 letters;
- , linear groups of a 0-dimensional vector space;
- , linear groups of a 1-dimensional vector space
- and many others.
Spheres
- , the last being the group of units of the integers,
- circle group
- unit quaternions.
Spin groups
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:| Diagram | Dynkin classification | Lie algebra | Polytope |
| A1 = B1 = C1 | - | ||
| A2 = I2 | - | 2-simplex is regular 3-gon | |
| BC2 = I2 | 2-cube is 2-cross polytope is regular 4-gon | ||
| A1 × A1 = D2 | - | ||
| A3 = D3 | 3-simplex is 3-demihypercube |