Exceptional isomorphism
# 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:

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

### Alternating groups and symmetric groups

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

* $S_{3}\cong \operatorname {PSL} _{2}(2)\cong$ Dihedral group of order 6,
* $A_{4}\cong \operatorname {PSL} _{2}(3)\cong$ tetrahedral group,
* $S_{4}\cong \operatorname {PGL} _{2}(3)\cong \operatorname {PSL} _{2}(\mathbb {Z} /4)\cong$ full tetrahedral group $\cong$ octahedral group,
* $A_{5}\cong \operatorname {PSL} _{2}(4)\cong \operatorname {PSL} _{2}(5)\cong$ icosahedral group,
* $S_{5}\cong \operatorname {P\Gamma L} _{2}(4)\cong \operatorname {PGL} _{2}(5)$,
* $A_{6}\cong \operatorname {PSL} _{2}(9)\cong \operatorname {Sp} _{4}(2)',$
* $S_{6}\cong \operatorname {Sp} _{4}(2),$
* $A_{8}\cong \operatorname {PSL} _{4}(2)\cong \operatorname {O} _{6}^{+}(2)',$
* $S_{8}\cong \operatorname {O} _{6}^{+}(2).$

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

There are also some coincidences with symmetries of regular polyhedra: the alternating group $A_5$ agrees with the icosahedral group (itself an exceptional object), and the double cover of the alternating group $A_5$ 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:

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

### Spheres

The spheres $S^0,$ $S^1,$ and $S^3$ admit group structures, which can be described in many ways:

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

### Spin groups

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

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

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 |
| - | - | - | - |
| | $A_1$ = $B_1$ = $C_1$ | ${\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{3}\cong {\mathfrak {sp}}_{1}$ | - |
| $\cong$ | $A_2$ = $I_2(2)$ | - | 2-simplex is regular 3-gon (equilateral triangle) |
| | $BC_2$ = $I_2(4)$ | ${\mathfrak {so}}_{5}\cong {\mathfrak {sp}}_{2}$ | 2-cube is 2-cross polytope is regular 4-gon (square) |
| $\cong$ | $A_1$ × $A_1$ = $D_2$ | ${\mathfrak {sl}}_{2}\oplus {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{4}$ | - |
| $\cong$ | $A_3$ = $D_3$ | ${\mathfrak {sl}}_{4}\cong {\mathfrak {so}}_{6}$ | 3-simplex is 3-demihypercube (regular tetrahedron) |
