Binary tetrahedral group
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 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 binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, 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 rotations.)
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.
The binary tetrahedral group, denoted by 2T, fits into the short exact sequence
The binary tetrahedral group is the covering group of the tetrahedral group. Thinking of the tetrahedral group as the alternating group on four letters, we thus have the binary tetrahedral group as the covering group,
The binary tetrahedral group can be written as a semidirect product
where Q is the quaternion group consisting of the 8 Lipschitz units and Z3 is the cyclic group of order 3 generated by ω = −½(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
The group 2T has a presentation given by
Generators with these relations are given by
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
The rotational symmetry group of the n-simplex can be considered as the alternating group on letters, and the corresponding binary group is a 2-fold covering group. For all higher dimensions except and (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. It was first used in model building by Paul Frampton and Thomas Kephart in 1994. Evidence connecting models based on the binary tetrahedral group to the real world is a goal for the Large Hadron Collider.
See also 
- binary polyhedral group
- binary cyclic group
- binary dihedral group
- binary octahedral group
- binary icosahedral group
- Case, E.M.; Robert Karplus, C.N. Yang (1956). "Strange Particles and the Conservation of Isotopic Spin". Physical Review 101: 874–876. doi:10.1103/PhysRev.101.874.
- Frampton, Paul H.; Thomas W. Kephart (1995). "Simple Nonabelian Finite Flavor Groups and Fermion Masses". International Journal of Modern Physics A10: 4689–4704 preprint hep–ph/9409330.
- 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. and 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