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

• [math]\displaystyle{ \operatorname{PSL}_2(4) \cong \operatorname{PSL}_2(5) \cong A_5, }[/math] the smallest non-abelian simple group (order 60) – icosahedral symmetry;
• [math]\displaystyle{ \operatorname{PSL}_2(7) \cong \operatorname{PSL}_3(2), }[/math] the second-smallest non-abelian simple group (order 168) – PSL(2,7);
• [math]\displaystyle{ \operatorname{PSL}_2(9) \cong A_6; }[/math]
• [math]\displaystyle{ \operatorname{PSL}_4(2) \cong A_8; }[/math]
• [math]\displaystyle{ \operatorname{PSU}_4(2) \cong \operatorname{PSp}_4(3), }[/math] 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:

• [math]\displaystyle{ S_3 \cong \operatorname{PSL}_2(2) \cong {} }[/math] Dihedral group of order 6,
• [math]\displaystyle{ A_4 \cong \operatorname{PSL}_2(3) \cong {} }[/math] tetrahedral group,
• [math]\displaystyle{ S_4 \cong \operatorname{PGL}_2(3) \cong \operatorname{PSL}_2(\mathbb{Z}/4) \cong {} }[/math] full tetrahedral group [math]\displaystyle{ \cong }[/math] octahedral group,
• [math]\displaystyle{ A_5 \cong \operatorname{PSL}_2(4) \cong \operatorname{PSL}_2(5) \cong {} }[/math] icosahedral group,
• [math]\displaystyle{ S_5 \cong \operatorname{P\Gamma L}_2(4) \cong \operatorname{PGL}_2(5) }[/math],
• [math]\displaystyle{ A_6 \cong \operatorname{PSL}_2(9) \cong \operatorname{Sp}_4(2)', }[/math]
• [math]\displaystyle{ S_6 \cong \operatorname{Sp}_4(2), }[/math]
• [math]\displaystyle{ A_8 \cong \operatorname{PSL}_4(2) \cong \operatorname{O}_6^+(2)', }[/math]
• [math]\displaystyle{ S_8 \cong \operatorname{O}_6^+(2). }[/math]

These can all be explained in a systematic way by using linear algebra (and the action of [math]\displaystyle{ S_n }[/math] on affine [math]\displaystyle{ n }[/math]-space) to define the isomorphism going from the right side to the left side. (The above isomorphisms for [math]\displaystyle{ A_8 }[/math] and [math]\displaystyle{ S_8 }[/math] are linked via the exceptional isomorphism [math]\displaystyle{ \operatorname{SL}_4/\mu_2 \cong \operatorname{SO}_6 }[/math].)

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:

• [math]\displaystyle{ C_1 }[/math], the cyclic group of order 1;
• [math]\displaystyle{ A_0 \cong A_1 \cong A_2 }[/math], the alternating group on 0, 1, or 2 letters;
• [math]\displaystyle{ S_0 \cong S_1 }[/math], the symmetric group on 0 or 1 letters;
• [math]\displaystyle{ \operatorname{GL}(0,\mathbb K) \cong \operatorname{SL}(0,\mathbb K) \cong \operatorname{PGL}(0,\mathbb K) \cong \operatorname{PSL}(0,\mathbb K) }[/math], linear groups of a 0-dimensional vector space;
• [math]\displaystyle{ \operatorname{SL}(1,\mathbb K) \cong \operatorname{PGL}(1,\mathbb K) \cong \operatorname{PSL}(1,\mathbb K) }[/math], 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:

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

Spin groups

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

• [math]\displaystyle{ \operatorname{Spin}(4) \cong \operatorname{Sp}(1) \times \operatorname{Sp}(1) \cong \operatorname{SU}(2) \times \operatorname{SU}(2) }[/math]
• [math]\displaystyle{ \operatorname{Spin}(5) \cong \operatorname{Sp}(2) }[/math]
• [math]\displaystyle{ \operatorname{Spin}(6) \cong \operatorname{SU}(4) }[/math]

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	[math]\displaystyle{ \mathfrak{sl}_2 \cong \mathfrak{so}_3 \cong \mathfrak{sp}_1 }[/math]	-
[math]\displaystyle{ \cong }[/math]	A2 = I2(2)	-	2-simplex is regular 3-gon (equilateral triangle)
	BC2 = I2(4)	[math]\displaystyle{ \mathfrak{so}_5 \cong \mathfrak{sp}_2 }[/math]	2-cube is 2-cross polytope is regular 4-gon (square)
[math]\displaystyle{ \cong }[/math]	A1 × A1 = D2	[math]\displaystyle{ \mathfrak{sl}_2 \oplus \mathfrak{sl}_2 \cong \mathfrak{so}_{4} }[/math]	-
[math]\displaystyle{ \cong }[/math]	A3 = D3	[math]\displaystyle{ \mathfrak{sl}_4 \cong \mathfrak{so}_6 }[/math]	3-simplex is 3-demihypercube (regular tetrahedron)

See also

• Exceptional object
• Mathematical coincidence, for numerical coincidences