Exceptional isomorphism - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...

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.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). These coincidences are at times considered a matter of trivia, but in other respects they can give rise to other phenomena, notably exceptional object

Exceptional object

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects as well as a finite number of exceptions that don't fit into any series. These are known as exceptional...

s. In the below, coincidences are listed in all places they occur.

Finite simple groups

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

the smallest non-abelian simple group (order 60);

the second-smallest non-abelian simple group (order 168) – PSL(2,7)

PSL(2,7)

In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane...

;

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

have a number of exceptional isomorphisms:

the alternating group on five elements, or equivalently the icosahedral group;

the symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

on five elements;

the double cover of the alternating group A₅

Covering groups of the alternating and symmetric groups

In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups...

, or equivalently the binary icosahedral group.

Alternating groups and symmetric groups

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

These can all be explained in a systematic way by using linear algebra (and the action of

on affine

-space)
to define the isomorphism going from the right side to the left side. (The above isomorphisms for

and

are linked via the exceptional isomorphism

.)
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

Covering groups of the alternating and symmetric groups

of the alternating group A5 is the binary icosahedral group.

Trivial group

The trivial group

Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...

arises in numerous ways; families often start with the trivial group which is discarded. For instance, it is:

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), groups of a 0-dimensional vector space;
SL(1,K) = PGL(1,K) = PSL(1,K), groups of a 1-dimensional vector space;
and many others.

Cyclic groups

Cyclic groups of small order especially arise in various ways, for instance:

C2 = ±1 = S⁰ (real numbers of unit norm) = O(1) = Spin(1) = Z^* (group of units of the integers)

Spheres

The spheres S⁰, S¹, and S³ admit group structures, which arise in various ways:

S⁰=O(1)=Spin(1) (this last properly a double cover);
S¹=SO(2)=U(1)=Spin(2) (this last properly a double cover; arising from Real/Complex structures);
S³=Spin(3)=SU(2)=Sp(1) (arising from Real/Complex/Quaternionic structures, respectively)

Coxeter groups

There are some exceptional isomorphisms of Coxeter diagrams, 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 algebras, visible as isomorphisms of the corresponding Dynkin diagrams:

Trivially, A0 = B0 = C0 = D0
A1 = B1 = C1 (= D1?)
B2 = C2
D2 = A1 × A1; note that these are disconnected, but part of the D-series
A3 = D3
A4 = E4; the E-series usually starts at 6, but can be started at 4, yielding isomorphisms
D5 = E5

Spin(1) = O(1)

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

Spin(2) = U(1)

Unitary group

In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...

= SO(2)

Spin(3) = Sp(1)

Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...

= SU(2)

Special unitary group

The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

Spin(4) = Sp(1)

Symplectic group

× Sp(1)

Symplectic group

Spin(5) = Sp(2)

Symplectic group

Spin(6) = SU(4)

Special unitary group

The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

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