Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members a_i and b_j of two families, usually infinite, of mathematical objects, which is incidental, in that it is not an instance of a general 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 consequential phenomena, such as exceptional objects.^[1] In the following, coincidences are organized according to the structures where 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:^[2]

PSL₂(4) ≅ PSL₂(5) ≅ A₅, the smallest non-abelian simple group (order 60);
PSL₂(7) ≅ PSL₃(2), the second-smallest non-abelian simple group (order 168) – PSL(2,7);
PSL₂(9) ≅ A₆;
PSL₄(2) ≅ A₈;
PSU₄(2) ≅ PSp₄(3), between a projective special unitary group and a projective symplectic group.

Alternating groups and symmetric groups

There are coincidences between symmetric/alternating groups and small groups of Lie type/polyhedral groups:^[3]

S₃ ≅ PSL₂(2) ≅ dihedral group of order 6,
A₄ ≅ PSL₂(3),
S₄ ≅ PGL₂(3) ≅ PSL₂(Z / 4),
A₅ ≅ PSL₂(4) ≅ PSL₂(5),
S₅ ≅ PΓL₂(4) ≅ PGL₂(5),
A₆ ≅ PSL₂(9) ≅ Sp₄(2)′,
S₆ ≅ Sp₄(2),
A₈ ≅ PSL₄(2) ≅ O⁺
₆(2)′,
S₈ ≅ O⁺
₆(2).

These can all be explained in a systematic way by using linear algebra (and the action of S_n on affine nspace) to define the isomorphism going from the right side to the left side. (The above isomorphisms for A₈ and S₈ are linked via the exceptional isomorphism SL₄ / μ₂ ≅ SO₆.)

There are also some coincidences with symmetries of regular polyhedra: the alternating group A₅ agrees with the chiral icosahedral group (itself an exceptional object), 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:

C₁, the cyclic group of order 1;
A₀ ≅ A₁ ≅ A₂, the alternating group on 0, 1, or 2 letters;
S₀ ≅ S₁, the symmetric group on 0 or 1 letters;
GL(0, K) ≅ SL(0, K) ≅ PGL(0, K) ≅ PSL(0, K), linear groups of a 0-dimensional vector space;
SL(1, K) ≅ PGL(1, K) ≅ PSL(1, K), linear groups of a 1-dimensional vector space
and many others.

Spheres

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

S⁰ ≅ Spin(1) ≅ O(1) ≅ (Z / 2Z)⁺ ≅ Z^×, the last being the group of units of the integers;
S¹ ≅ Spin(2) ≅ SO(2) ≅ U(1) ≅ R / Z ≅ circle group;
S³ ≅ Spin(3) ≅ SU(2) ≅ Sp(1) ≅ unit quaternions.

Spin groups

In addition to Spin(1), Spin(2) and Spin(3) above, there are isomorphisms for higher dimensional spin groups:

Spin(4) ≅ Sp(1) × Sp(1) ≅ SU(2) × SU(2)
Spin(5) ≅ Sp(2)
Spin(6) ≅ SU(4)

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:

Diagram	Dynkin classification	Lie algebra	Polytope
	A₁ = B₁ = C₁	${\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{3}\cong {\mathfrak {sp}}_{1}$	—
$\cong$	A₂ = I₂(2)	—	2-simplex is regular 3-gon (equilateral triangle)
	BC₂ = I₂(4)	${\mathfrak {so}}_{5}\cong {\mathfrak {sp}}_{2}$	2-cube is 2-cross polytope is regular 4-gon (square)
$\cong$	A₁ × A₁ = D₂	${\mathfrak {sl}}_{2}\oplus {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{4}$	—
$\cong$	A₃ = D₃	${\mathfrak {sl}}_{4}\cong {\mathfrak {so}}_{6}$	3-simplex is 3-demihypercube (regular tetrahedron)

Notes

^ 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

^ ^a ^b Wilson 2009, Chapter 1: Introduction
^ Wilson 2009, Chapter 1: Introduction
^ Wilson 2009, Chapter 3

Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012[www.maths.qmul.ac.uk/~raw/fsgs.html 2007 preprint]{{citation}}: CS1 maint: postscript (link)

[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).

[FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]-2] Wilson 2009, Chapter 1: Introduction

[FOOTNOTEWilson2009[httpswebarchiveorgweb20070824011953httpswwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]-3] Wilson 2009, Chapter 1: Introduction

[FOOTNOTEWilson2009Chapter_3-4] Wilson 2009, Chapter 3

[note 1]

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