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{{expert-subject|mathematics|date=December 2009}}

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.<ref group="note">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).</ref> These coincidences are at times considered a matter of trivia,<ref name="raw"/> but in other respects they can give rise to other phenomena, notably [[exceptional object]]s.<ref name="raw"/> In the below, coincidences are listed in all places they occur.

== Groups ==
=== Finite simple groups ===
The exceptional isomorphisms between the series of [[finite simple group]]s mostly involve [[projective special linear group]]s and [[alternating group]]s, and are:<ref name="raw">{{Citation | last1=Wilson | first1=Robert A. | authorlink = Robert Arnott Wilson | title=The finite simple groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | year=2009 |chapter = Chapter 1: Introduction |chapterurl=http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps | postscript =, [http://www.maths.qmul.ac.uk/~raw/fsgs.html 2007 preprint]; Chapter {{doi|10.1007/978-1-84800-988-2_1}}. | zbl=05622792 | volume=251}}</ref>
*<math>L_2(4) \cong L_2(5) \cong A_5,</math> the smallest non-abelian simple group (order 60);
*<math>L_2(7) \cong L_3(2),</math> the second-smallest non-abelian simple group (order 168) – [[PSL(2,7)]];
*<math>L_2(9) \cong A_6,</math>
*<math>L_4(2) \cong A_8,</math>
*<math>\operatorname{PSU}_4(2) \cong \operatorname{PSp}_4(3),</math> between a [[projective special orthogonal group]] and a [[projective symplectic group]].

=== Groups of Lie type ===
In addition to the aforementioned, there are some isomorphisms involving SL, PSL, GL, PGL, and the natural maps between these. For example, the groups over <math>\mathbf{F}_5</math> have a number of exceptional isomorphisms:
*<math>\operatorname{PSL}(2,5) \cong A_5 \cong I,</math> the alternating group on five elements, or equivalently the [[icosahedral group]];
*<math>\operatorname{PGL}(2,5) \cong S_5,</math> the [[symmetric group]] on five elements;
*<math>\operatorname{SL}(2,5) \cong 2\cdot A_5 \cong 2I,</math> the [[covering groups of the alternating and symmetric groups|double cover of the alternating group ''A''5]], or equivalently the [[binary icosahedral group]].

=== Alternating groups and symmetric groups ===
[[File:Compound of five tetrahedra.png|thumb|The [[compound of five tetrahedra]] expresses the exceptional isomorphism between the icosahedral group and the alternating group on five letters.]]
There are coincidences between alternating groups and small groups of Lie type:
*<math>L_2(4) \cong L_2(5) \cong A_5,</math>
*<math>L_2(9) \cong A_6,</math>
*<math>L_4(2) \cong A_8,</math>
*<math>O_6(2) \cong S_8.</math>
These can all be explained in a systematic way by using linear algebra (and the action of <math>S_n</math> on affine <math>n</math>-space)
to define the isomorphism going from the right side to the left side. (The above isomorphisms for <math>A_8</math> and <math>S_8</math> are linked via the exceptional isomorphism <math>SL_4/\mu_2 \cong 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 [[covering groups of the alternating and symmetric groups|double cover]] of the alternating group A5 is the [[binary icosahedral group]].

=== Cyclic groups ===
Cyclic groups of small order especially arise in various ways, for instance:
* <math> C_2 \cong \{\pm1\} \cong \operatorname{O}(1) \cong \operatorname{Spin}(1) \cong \mathbb Z^*</math>, the last being the group of units of the integers

=== Spheres ===
The spheres S0, S1, and S3 admit group structures, which arise in various ways:
* <math> S^0\cong\operatorname{O}(1) </math>,
* <math> S^1\cong\operatorname{SO}(2)\cong\operatorname{U}(1)\cong\operatorname{Spin}(2) </math>,
* <math> S^3\cong\operatorname{Spin}(3)\cong\operatorname{SU}(2)\cong\operatorname{Sp}(1) </math>.

=== Coxeter groups ===
[[File:Dynkin Diagram Isomorphisms.svg|thumb|upright|The exceptional isomorphisms of connected [[Dynkin diagram]]s.]]
There are some exceptional isomorphisms of [[Coxeter diagram]]s, yielding isomorphisms of the corresponding Coxeter groups and of polytopes realizing the symmetries. These are:
* A2 = I2(2) (2-simplex is regular 3-gon/triangle);
* BC2 = I2(4) (2-cube (square) = 2-cross-polytope (diamond) = regular 4-gon)
* A3 = D3 (3-simplex (tetrahedron) is 3-demihypercube (demicube), as per diagram)
* A1 = B1 = C1 (= D1?)
* D2 = A1 × A1
* A4 = E4
* D5 = E5

Closely related ones occur in Lie theory for Dynkin diagrams.

== Lie theory ==

In low dimensions, there are isomorphisms among the classical Lie algebras and classical Lie groups called ''accidental isomorphisms''. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups, due to low dimensional isomorphisms between the [[root systems]] of the different families of [[simple Lie algebra]]s, visible as isomorphisms of the corresponding Dynkin diagrams:
* Trivially, A0 = B0 = C0 = D0
* A1 = B1 = C1 , or <math>\mathfrak{sl}_2 \cong \mathfrak{so}_3 \cong \mathfrak{sp}_1 </math>
* B2 = C2, or <math>\mathfrak{so}_5 \cong \mathfrak{sp}_2 </math>
* D2 = A1 × A1, or <math>\mathfrak{so}_{4} \cong \mathfrak{sl}_2 \oplus \mathfrak{sl}_2 </math>; note that these are disconnected, but part of the D-series
* A3 = D3 <math>\mathfrak{sl}_4 \cong \mathfrak{so}_6</math>
* A4 = E4; the E-series usually starts at 6, but can be started at 4, yielding isomorphisms
* D5 = E5

:Spin(1) = [[Orthogonal group|O(1)]]
:Spin(2) = [[Unitary group|U(1)]] = [[Special orthogonal group|SO(2)]]
:Spin(3) = [[Symplectic group|Sp(1)]] = [[Special unitary group|SU(2)]]
:Spin(4) = [[Symplectic group|Sp(1)]] × [[Symplectic group|Sp(1)]]
:Spin(5) = [[Symplectic group|Sp(2)]]
:Spin(6) = [[Special unitary group|SU(4)]]

== See also ==
* [[Exceptional object]]
* [[Mathematical coincidence]], for numerical coincidences

== References ==
{{reflist|group=note}}

== References ==
{{reflist}}
{{refbegin}}
{{refend}}

[[Category:Mathematical relations]]

Radial function - Revision history

en>Frietjes: Clean up duplicate template arguments using findargdups

en>Brad7777: added Category:Types of functions using HotCat