Rotational diffusion

In mathematics, the binary tetrahedral group, denoted 2T or Template:Langle2,3,3Template:Rangle 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 binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism $\operatorname {Spin} (3)\cong \operatorname {Sp} (1)$ 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

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

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

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, $T\cong A_{4},$ we thus have the binary tetrahedral group as the covering group, $2T\cong {\widehat {A_{4}}}.$

The center of 2T is the subgroup {±1}. The outer automorphism group is trivial, so that the inner automorphism group is isomorphic to the full automorphism group, which is the tetrahedral group T.

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

$2T=Q\rtimes \mathbb {Z} _{3}$

where Template:Mvar is the quaternion group consisting of the 8 Lipschitz units and Z₃ is the cyclic group of order 3 generated by Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.. The group Z₃ acts on the normal subgroup Template:Mvar by conjugation. Conjugation by Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) – the group of all Template:Gaps 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

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

Subgroups

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
• binary dihedral group
• binary octahedral group
• binary icosahedral group

Notes

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References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 6.5 The binary polyhedral groups, p. 68