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, 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

Finite simple groups

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

• ${\displaystyle \operatorname {PSL} _{2}(4)\cong \operatorname {PSL} _{2}(5)\cong A_{5},}$ the smallest non-abelian simple group (order 60) – icosahedral symmetry;
• ${\displaystyle \operatorname {PSL} _{2}(7)\cong \operatorname {PSL} _{3}(2),}$ the second-smallest non-abelian simple group (order 168) – PSL(2,7);
• ${\displaystyle \operatorname {PSL} _{2}(9)\cong A_{6},}$
• ${\displaystyle \operatorname {PSL} _{4}(2)\cong A_{8},}$
• ${\displaystyle \operatorname {PSU} _{4}(2)\cong \operatorname {PSp} _{4}(3),}$ between a projective special orthogonal group and a projective symplectic group.

Alternating groups and symmetric groups

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]

• ${\displaystyle S_{3}\cong \operatorname {PSL} _{2}(2),}$
• ${\displaystyle A_{4}\cong \operatorname {PSL} _{2}(3)\cong }$ tetrahedral group,
• ${\displaystyle S_{4}\cong }$ full tetrahedral group ${\displaystyle \cong }$ octahedral group,
• ${\displaystyle A_{5}\cong \operatorname {PSL} _{2}(4)\cong \operatorname {PSL} _{2}(5)\cong }$ icosahedral group,
• ${\displaystyle A_{6}\cong \operatorname {PSL} _{2}(9)\cong \operatorname {Sp} _{4}(2)',}$
• ${\displaystyle S_{6}\cong \operatorname {Sp} _{4}(2),}$
• ${\displaystyle A_{8}\cong \operatorname {PSL} _{4}(2)\cong \operatorname {O} _{6}^{+}(2)',}$
• ${\displaystyle S_{8}\cong \operatorname {O} _{6}^{+}(2).}$

These can all be explained in a systematic way by using linear algebra (and the action of ${\displaystyle S_{n}}$ on affine ${\displaystyle n}$-space) to define the isomorphism going from the right side to the left side. (The above isomorphisms for ${\displaystyle A_{8}}$ and ${\displaystyle S_{8}}$ are linked via the exceptional isomorphism ${\displaystyle \operatorname {SL} _{4}/\mu _{2}\cong \operatorname {SO} _{6}}$.) 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

The trivial group arises in numerous ways. The trivial group is often omitted from the beginning of a classical family. For instance:

• ${\displaystyle C_{1}}$, the cyclic group of order 1;
• ${\displaystyle A_{0}\cong A_{1}\cong A_{2}}$, the alternating group on 0, 1, or 2 letters;
• ${\displaystyle S_{0}\cong S_{1}}$, the symmetric group on 0 or 1 letters;
• ${\displaystyle \operatorname {GL} (0,\mathbb {K} )\cong \operatorname {SL} (0,\mathbb {K} )\cong \operatorname {PGL} (0,\mathbb {K} )\cong \operatorname {PSL} (0,\mathbb {K} )}$, linear groups of a 0-dimensional vector space;
• ${\displaystyle \operatorname {SL} (1,\mathbb {K} )\cong \operatorname {PGL} (1,\mathbb {K} )\cong \operatorname {PSL} (1,\mathbb {K} )}$, 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:

• ${\displaystyle S^{0}\cong \operatorname {Spin} (1)\cong \operatorname {O} (1)\cong \mathbb {Z} /2\mathbb {Z} \cong \mathbb {Z} ^{\times }}$, the last being the group of units of the integers ,
• ${\displaystyle S^{1}\cong \operatorname {Spin} (2)\cong \operatorname {SO} (2)\cong \operatorname {U} (1)\cong \mathbb {R} /\mathbb {Z} \cong }$ circle group
• ${\displaystyle S^{3}\cong \operatorname {Spin} (3)\cong \operatorname {SU} (2)\cong \operatorname {Sp} (1)\cong }$ unit quaternions.

Spin groups

In addition to ${\displaystyle \operatorname {Spin} (1)}$, ${\displaystyle \operatorname {Spin} (2)}$ and ${\displaystyle \operatorname {Spin} (3)}$ above, there are isomorphisms for higher dimensional spin groups:

• ${\displaystyle \operatorname {Spin} (4)\cong \operatorname {Sp} (1)\times \operatorname {Sp} (1)\cong \operatorname {SU} (2)\times \operatorname {SU} (2)}$
• ${\displaystyle \operatorname {Spin} (5)\cong \operatorname {Sp} (2)}$
• ${\displaystyle \operatorname {Spin} (6)\cong \operatorname {SU} (4)}$

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

See also

• Exceptional object
• Mathematical coincidence, for numerical coincidences

Notes

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

1. 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