Exceptional isomorphism

[note 1] These coincidences are at times considered a matter of trivia,[1] but in other respects they can give rise to consequential phenomena, such as exceptional objects.

[1] In the following, coincidences are organized according to the structures where they occur.

The trivial group arises in numerous ways.

The trivial group is often omitted from the beginning of a classical family.

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.