Binary tetrahedral group

The regular complex polytope, 3{6}2 or or , represents the Cayley diagram for the binary tetrahedral group, with each red and blue triangle a directed subgraph.

In mathematics, the binary tetrahedral group, denoted 2T or ⟨2,3,3⟩, is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C. Shephard or 3[3]3 and by Coxeter, is isomorphic to the binary tetrahedral group.

The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)

Elements

Cayley graph of SL(2,3)

Symmetry projections

8-fold	12-fold
24 quaternion elements: 1 order-1: 1 1 order-2: -1 6 order-4: ±i, ±j, ±k 8 order-6: (+1±i±j±k)/2 8 order-3: (-1±i±j±k)/2.

Explicitly, the binary tetrahedral group is given as the group of units in the ring of Hurwitz integers. There are 24 such units given by

$\{\pm 1,\pm i,\pm j,\pm k,{\tfrac {1}{2}}(\pm 1\pm i\pm j\pm k)\}$

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 of these 24 elements in 4-dimensional space form a convex regular 4-polytope called the 24-cell.

Properties

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

$1\to \{\pm 1\}\to 2\mathrm {T} \to \mathrm {T} \to 1.$

This sequence does not split, meaning that 2T is not a semidirect product of {±1} by T. In fact, there is no subgroup of 2T isomorphic to T.

The binary tetrahedral group is the covering group of the tetrahedral group. Thinking of the tetrahedral group as the alternating group on four letters, T ≅ A₄, we thus have the binary tetrahedral group as the covering group, 2T ≅ ${\widehat {\mathrm {A} _{4}}}$.

The center of 2T is the subgroup {±1}. The inner automorphism group is isomorphic to A₄, and the full automorphism group is isomorphic to S₄.

Left multiplication by −ω, an order-6 element: look at gray, blue, purple, and orange balls and arrows that constitute 4 orbits (two arrows are not depicted). ω itself is the bottommost ball: ω = (−ω)(−1) = (−ω)⁴

The binary tetrahedral group can be written as a semidirect product

$2\mathrm {T} =\mathrm {Q} \rtimes \mathrm {C} _{3}$

where Q is the quaternion group consisting of the 8 Lipschitz units and C₃ is the cyclic group of order 3 generated by ω = −⁠1/2⁠(1 + i + j + k). The group Z₃ acts on the normal subgroup Q by conjugation. Conjugation by ω is the automorphism of Q that cyclically rotates i, j, and k.

One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) – the group of all 2 × 2 matrices over the finite field F₃ with unit determinant, with this isomorphism covering the isomorphism of the projective special linear group PSL(2,3) with the alternating group A₄.

Presentation

The group 2T has a presentation 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

$r=i\qquad s={\tfrac {1}{2}}(1+i+j+k)\qquad t={\tfrac {1}{2}}(1+i+j-k),$

with $r^{2}=s^{3}=t^{3}=-1$.

A Cayley Table 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 $e_{14}=s$ and $e_{18}=t$).

Subgroups

The binary tetrahedral group, 2T=<3,3,2>, has 2 primary subgroups:
• Quaternion group, Q=<2,2,2>, index 3
• Cyclic group Z6=<3>, index 4.

The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2T of index 3. This group and the center {±1} are the only nontrivial normal subgroups.

All other subgroups of 2T are cyclic groups 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 (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 alternating group on n + 1 points, A_n+1, and the corresponding binary group is a 2-fold covering group. For all higher dimensions except A₆ and A₇ (corresponding to the 5-dimensional and 6-dimensional simplexes), this binary group is the covering group (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 in 1956 by Chen Ning Yang 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.

See also

• Binary polyhedral group
• Binary cyclic group, ⟨n⟩, order 2n
• Binary dihedral group, ⟨2,2,n⟩, order 4n
• Binary octahedral group, 2O = ⟨2,3,4⟩, order 48
• Binary icosahedral group, 2I = ⟨2,3,5⟩, order 120

References

• Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters, Ltd. ISBN 1-56881-134-9.
• Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups, 4th edition. New York: Springer-Verlag. ISBN 0-387-09212-9. 6.5 The binary polyhedral groups, p. 68
• "Special linear group SL(2,3)".