Exceptional isomorphism

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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.[note 1] These coincidences are at times considered a matter of trivia,[1] but in other respects they can give rise to other phenomena, notably exceptional objects.[1] In the following, coincidences are listed wherever they occur.

Groups[edit]

Finite simple groups[edit]

The exceptional isomorphisms between the series of finite simple groups mostly involve projective special linear groups and alternating groups, and are:[1]

  • the smallest non-abelian simple group (order 60) – icosahedral symmetry;
  • the second-smallest non-abelian simple group (order 168) – PSL(2,7);
  • between a projective special orthogonal group and a projective symplectic group.

Alternating groups and symmetric groups[edit]

The compound of five tetrahedra expresses the exceptional isomorphism between the icosahedral group and the alternating group on five letters.

There are coincidences between symmetric/alternating groups and small groups of Lie type/polyhedral groups:[2]

  • tetrahedral group,
  • full tetrahedral group octahedral group,
  • icosahedral group,

These can all be explained in a systematic way by using linear algebra (and the action of on affine -space) to define the isomorphism going from the right side to the left side. (The above isomorphisms for and are linked via the exceptional isomorphism .) There are also some coincidences with symmetries of regular polyhedra: the alternating group A5 agrees with the icosahedral group (itself an exceptional object), and the double cover of the alternating group A5 is the binary icosahedral group.

Trivial group[edit]

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[edit]

The spheres S0, S1, and S3 admit group structures, which can be described in many ways:

  • , the last being the group of units of the integers ,
  • circle group
  • unit quaternions.

Spin groups[edit]

In addition to , and above, there are isomorphisms for higher dimensional spin groups:

Also, Spin(8) has an exceptional order 3 triality automorphism

Coxeter–Dynkin diagrams[edit]

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
CDel node.png A1 = B1 = C1 -
CDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3x.pngCDel node.png A2 = I2(2) - 2-simplex is regular 3-gon (equilateral triangle)
CDel node.pngCDel 4.pngCDel node.png BC2 = I2(4) 2-cube is 2-cross polytope is regular 4-gon (square)
CDel node.png CDel node.png CDel nodes.png A1 × A1 = D2 -
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel split1.pngCDel nodes.png A3 = D3 3-simplex is 3-demihypercube (regular tetrahedron)

See also[edit]

Notes[edit]

  1. ^ Because these series of objects are presented differently, they are not identical objects (do not have identical descriptions), but turn out to describe the same object, hence one refers to this as an isomorphism, not an equality (identity).

References[edit]

  1. ^ a b c Wilson, Robert A. (2009), "Chapter 1: Introduction", The finite simple groups, Graduate Texts in Mathematics 251, 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012, 2007 preprint; Chapter doi:10.1007/978-1-84800-988-2_1.
  2. ^ Wilson, Robert A. (2009), Chapter 3