mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the binary tetrahedral group, denoted 2T or ,Coxeter&Moser: Generators and Relations for discrete groups: : R^l = S^m = Tⁿ = RST is a certain

nonabelian group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ' ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

24. It is an

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values that ...

of the

tetrahedral group 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...

T or (2,3,3) of order 12 by a

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

of order 2, and is the

preimage In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each ...

of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the

special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

by the

spin group In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \o ...

. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The

complex reflection group In mathematics, a complex reflection group is a Group (mathematics), finite group acting on a finite-dimensional vector space, finite-dimensional complex numbers, complex vector space that is generated by complex reflections: non-trivial elements t ...

named 3(24)3 by G.C. Shephard or 3 and by

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

, is isomorphic to the binary tetrahedral group. The binary tetrahedral group is most easily described concretely as a discrete subgroup of the

unit quaternion In mathematics, a versor is a quaternion of norm one, also known as a unit quaternion. Each versor has the form :u = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi where the r2 = −1 condition means that r is ...

s, under the isomorphism , where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on

quaternions and spatial rotation unit vector, Unit quaternions, known as versor, ''versors'', provide a convenient mathematics, mathematical notation for representing spatial Orientation (geometry), orientations and rotations of elements in three dimensional space. Specifically, th ...

s.)

Elements

Explicitly, the binary tetrahedral group is given as the

group of units In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the ele ...

in the

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

Hurwitz integer In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are ''either'' all integers ''or'' all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz ...

s. There are 24 such units given by :

\

with all possible sign combinations. All 24 units have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of these 24 elements in 4-dimensional space form a

convex regular 4-polytope In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six co ...

called the

24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...

Properties

The binary tetrahedral group, denoted by 2T, fits into the

short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...

1\to\\to 2\mathrm\to \mathrm \to 1.

This sequence does not

split Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, enter ...

, meaning that 2T is ''not'' a

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...

of by T. In fact, there is no subgroup of 2T isomorphic to T. The binary tetrahedral group is the

covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous (topology), continuous group homomorphism. The map ''p'' is called the c ...

of the tetrahedral group. Thinking of the tetrahedral group as the

alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...

on four letters, , we thus have the binary tetrahedral group as the covering group, The center of 2T is the subgroup . The

inner automorphism group In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via operations from within the group itself, ...

is isomorphic to A₄, and the full

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

is isomorphic to S₄. The binary tetrahedral group can be written as a

2\mathrm=\mathrm\rtimes\mathrm_3

where Q is the

quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a nonabelian group, non-abelian group (mathematics), group of Group order, order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. ...

consisting of the 8 Lipschitz units and C₃ is the

of order 3 generated by . The group Z₃ acts on the normal subgroup Q by

conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change o ...

. Conjugation by is the automorphism of Q that cyclically rotates , , and . One can show that the binary tetrahedral group is isomorphic to the

special linear group In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...

SL(2,3) – the group of all matrices over the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

F₃ with unit determinant, with this isomorphism covering the isomorphism of the

projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...

PSL(2,3) with the alternating group A₄.

Presentation

The group 2T has a

presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

given by :

\langle r,s,t \mid r^2 = s^3 = t^3 = rst \rangle

or equivalently, :

\langle s,t \mid (st)^2 = s^3 = t^3 \rangle.

Generators with these relations are given by :

r = i \qquad s = \tfrac(1+i+j+k) \qquad t = \tfrac(1+i+j-k),

with

r^2 = s^3 = t^3 = -1

. A

Cayley Table Named after the 19th-century United Kingdom, British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an additi ...

with these properties, elements ordered by GAP, is

  1  2  r  4 -1  6  7  8  9 10 11 12 13  s 15 16 17  t 19 20 21 22 23 24
  2  6  7  8  9  1 13  s 15 16 17  t  r  4 -1 20 21 22 23 10 11 12 24 19
  r  8 -1 10 11 20 23  9  t 12  1 19  s 21 24  7 16  2  4 15 22 13 17  6
  4 16 19 -1 12 13  8 17 23  r 10  1 15 20 21  9  t  7 11 22  6 24  2  s
 -1  9 11 12  1 15 17  t  2 19  r  4 21 22  6 23  7  8 10 24 13  s 16 20
  6  1 13  s 15  2  r  4 -1 20 21 22  7  8  9 10 11 12 24 16 17  t 19 23
  7  s  9 16 17 10 24 15 22  t  2 23  4 11 19 13 20  6  8 -1 12  r 21  1
  8 20 23  9  t  r  s 21 24  7 16  2 -1 10 11 15 22 13 17 12  1 19  6  4
  9 15 17  t  2 -1 21 22  6 23  7  8 11 12  1 24 13  s 16 19  r  4 20 10
 10  7  4 11 19  s  9 16 17 -1 12  r 24 15 22  t  2 23  1 13 20  6  8 21
 11  t  1 19  r 24 16  2  8  4 -1 10 22 13 20 17 23  9 12  6  s 21  7 15
 12 23 10  1  4 21  t  7 16 11 19 -1  6 24 13  2  8 17  r  s 15 20  9 22
 13  4 15 20 21 16 19 -1 12 22  6 24  8 17 23  r 10  1  s  9  t  7 11  2
  s 10 24 15 22  7  4 11 19 13 20  6  9 16 17 -1 12  r 21  t  2 23  1  8
 15 -1 21 22  6  9 11 12  1 24 13  s 17  t  2 19  r  4 20 23  7  8 10 16
 16 13  8 17 23  4 15 20 21  9  t  7 19 -1 12 22  6 24  2  r 10  1  s 11
 17 22  2 23  7 19 20  6  s  8  9 16 12  r 10 21 24 15  t  1  4 11 13 -1
  t 24 16  2  8 11 22 13 20 17 23  9  1 19  r  6  s 21  7  4 -1 10 15 12
 19 17 12  r 10 22  2 23  7  1  4 11 20  6  s  8  9 16 -1 21 24 15  t 13
 20  r  s 21 24  8 -1 10 11 15 22 13 23  9  t 12  1 19  6  7 16  2  4 17
 21 12  6 24 13 23 10  1  4  s 15 20  t  7 16 11 19 -1 22  2  8 17  r  9
 22 19 20  6  s 17 12  r 10 21 24 15  2 23  7  1  4 11 13  8  9 16 -1  t
 23 21  t  7 16 12  6 24 13  2  8 17 10  1  4  s 15 20  9 11 19 -1 22  r
 24 11 22 13 20  t  1 19  r  6  s 21 16  2  8  4 -1 10 15 17 23  9 12  7

There is 1 element of order 1 (element 1), one element of order 2 (

e_5=-1

), 8 elements of order 3, 6 elements of order 4 (including

e_3=r

), 8 elements of order 6 (which include

e_=s

and

e_=t

Subgroups

The

consisting of the 8 Lipschitz units forms a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

of 2T of

index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...

3. This group and the center are the only nontrivial normal subgroups. All other subgroups of 2T are

s generated by the various elements, with orders 3, 4, and 6.

Higher dimensions

Just as the tetrahedral group generalizes to the rotational symmetry group of the ''n''-

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

(as a subgroup of SO(''n'')), there is a corresponding higher binary group which is a 2-fold cover, coming from the cover Spin(''n'') → SO(''n''). The rotational symmetry group of the ''n''-simplex can be considered as the

on ''n'' + 1 points, A_''n''+1, and the corresponding binary group is a 2-fold

. For all higher dimensions except A₆ and A₇ (corresponding to the 5-dimensional and 6-dimensional simplexes), this binary group is the

(maximal cover) and is superperfect, but for dimensional 5 and 6 there is an additional exceptional 3-fold cover, and the binary groups are not superperfect.

Usage in theoretical physics

The binary tetrahedral group was used in the context of

Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...

in 1956 by

Chen Ning Yang Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge th ...

and others. It was first used in flavor physics model building by Paul Frampton and Thomas Kephart in 1994. In 2012 it was shown that a relation between two neutrino mixing angles, derived by using this binary tetrahedral flavor symmetry, agrees with experiment.

Notes

References

* * 6.5 The binary polyhedral groups, p. 68 *{{cite web, url=https://groupprops.subwiki.org/wiki/Special_linear_group:SL(2,3), title=Special linear group SL(2,3)

Tetrahedral In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...