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| In [[mathematics]], the '''binary tetrahedral group''', denoted 2''T'' or {{langle}}2,3,3{{rangle}} is a certain [[nonabelian group]] of [[order (group theory)|order]] 24. It is an [[group extension|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.
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| The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit [[quaternion]]s, under the isomorphism <math>\operatorname{Spin}(3) \cong \operatorname{Sp}(1)</math> where [[Sp(1)]] is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on [[quaternions and spatial rotation]]s.)
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| ==Elements==
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| Explicitly, the binary tetrahedral group is given as the [[group of units]] in the [[ring (mathematics)|ring]] of [[Hurwitz integer]]s. There are 24 such units given by
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| :<math>\{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1 \pm i \pm j \pm k)\}</math>
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| with all possible sign combinations.
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| 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]].
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| ==Properties==
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| The binary tetrahedral group, denoted by 2''T'', fits into the [[short exact sequence]]
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| :<math>1\to\{\pm 1\}\to 2T\to T \to 1.</math>
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| This sequence does not [[split exact sequence|split]], meaning that 2''T'' is ''not'' a [[semidirect product]] of {±1} by ''T''. In fact, there is no subgroup of 2''T'' isomorphic to ''T''.
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| The binary tetrahedral group is the [[covering groups of the alternating and symmetric groups|covering group]] of the tetrahedral group. Thinking of the tetrahedral group as the [[alternating group]] on four letters, <math>T \cong A_4,</math> we thus have the binary tetrahedral group as the covering group, <math>2T \cong \widehat{A_4}.</math> | |
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| The [[center of a group|center]] of 2''T'' 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''.
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| [[Image:Versor action on Hurwitz quaternions.svg|thumb|right|Left multiplication by −ω, an [[order (group theory)|order]]-6 element: look at gray, blue, purple, and orange balls and arrows that constitute 4 [[group action|orbits]] (two arrows are not depicted). ω itself is the bottommost ball: ω = (−ω)(−1) = (−ω)<sup>4</sup>]]
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| The binary tetrahedral group can be written as a [[semidirect product]]
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| :<math>2T=Q\rtimes\mathbb Z_3</math>
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| where {{mvar|Q}} is the [[quaternion group]] consisting of the 8 [[Lipschitz unit]]s and '''Z'''<sub>3</sub> is the [[cyclic group]] of order 3 generated by {{math|1=ω = −{{sfrac|1|2}}(1 + ''i'' + ''j'' + ''k'')}}. The group '''Z'''<sub>3</sub> acts on the normal subgroup {{mvar|Q}} by [[conjugation (group theory)|conjugation]]. Conjugation by {{math|ω}} is the automorphism of ''Q'' that cyclically rotates {{mvar|i}}, {{mvar|j}}, and {{mvar|k}}. | |
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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 {{gaps|2|×|2}} matrices over the [[finite field]] '''F'''<sub>3</sub> with unit determinant, with this isomorphism covering the isomorphism of the [[projective special linear group]] PSL(2,3) with the alternating group ''A''<sub>4</sub>.<!-- Is there any geometric meaning to this isomorphism? -->
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| ===Presentation===
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| The group 2''T'' has a [[group presentation|presentation]] given by
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| :<math>\langle r,s,t \mid r^2 = s^3 = t^3 = rst \rangle</math>
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| or equivalently,
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| :<math>\langle s,t \mid (st)^2 = s^3 = t^3 \rangle.</math>
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| Generators with these relations are given by
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| :<math>s = \tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{2}(1+i+j-k).</math>
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| ===Subgroups===
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| The [[quaternion group]] consisting of the 8 [[Lipschitz unit]]s forms a [[normal subgroup]] of 2''T'' of [[index (group theory)|index]] 3. This group and the center {±1} are the only nontrivial normal subgroups.
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| All other subgroups of 2''T'' are [[cyclic group]]s generated by the various elements, with orders 3, 4, and 6.
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| ==Higher dimensions==
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| 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'').
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| The rotational symmetry group of the ''n''-simplex can be considered as the [[alternating group]] on ''n'' + 1 points, ''A''<sub>''n''+1</sub>, and the corresponding binary group is a 2-fold [[covering groups of the alternating and symmetric groups|covering group]]. For all higher dimensions except ''A''<sub>6</sub> and ''A''<sub>7</sub> (corresponding to the 5-dimensional and 6-dimensional simplexes), this binary group is the [[universal perfect central extension|covering group]] (maximal cover) and is [[superperfect group|superperfect]], but for dimensional 5 and 6 there is an additional exceptional 3-fold cover, and the binary groups are not superperfect.
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| ==Usage in theoretical physics==
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| The binary tetrahedral group was used in the context of [[Yang-Mills theory]] in 1956 by [[Chen Ning Yang]] and others.<ref>{{cite journal
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| | last = Case
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| | first = E.M.
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| | coauthors = Robert Karplus, C.N. Yang
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| | title = Strange Particles and the Conservation of Isotopic Spin
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| | journal = Physical Review
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| | volume = 101
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| | pages = 874–876
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| | date = 1956
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| | doi = 10.1103/PhysRev.101.874
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| }}</ref>
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| It was first used in flavor physics model building by [[Paul Frampton]] and Thomas Kephart in 1994.<ref>{{cite journal
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| | last = Frampton
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| | first = Paul H.
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| | coauthors = Thomas W. Kephart
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| | title = Simple Nonabelian Finite Flavor Groups and Fermion Masses
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| | journal = International Journal of Modern Physics
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| | volume = A10
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| | pages = 4689–4704
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| |arxiv=hep-ph/9409330
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| | date = 1995
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| }}</ref>
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| In 2012 it was shown <ref>{{cite journal
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| | last = Eby
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| | first = David A.
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| | coauthors = Paul H. Frampton
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| | title = Nonzero theta(13)signals nonmaximal atmospheric neutrino mixing
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| | journal = Physical Review
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| | volume = D86
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| | pages = 117–304
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| |arxiv=1112.2675<!--hep-ph-->
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| | date = 2012
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| }}</ref> that a relation between two neutrino mixing angles,
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| derived
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| <ref>{{cite journal
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| | last = Eby
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| | first = David A.
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| | coauthors = Paul H. Frampton, Shinya Matsuzaki
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| | title = Predictions for neutrino mixing angles in a T′ Model
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| | journal = Physics Letters
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| | volume = B671
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| | pages = 386–390
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| |arxiv=0801.4899<!--hep-ph-->
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| | date = 2009
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| }}</ref>
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| by using this binary tetrahedral flavor symmetry, agrees with experiment.
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| ==See also==
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| *[[binary polyhedral group]]
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| *[[binary cyclic group]]
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| *[[binary dihedral group]]
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| *[[binary octahedral group]]
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| *[[binary icosahedral group]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book | first = John H. | last = Conway | coauthors = Smith, Derek A. | authorlink = John Horton Conway | title = On Quaternions and Octonions | publisher = AK Peters, Ltd | location = Natick, Massachusetts | year = 2003 | isbn = 1-56881-134-9}}
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| *{{cite book | author=Coxeter, H. S. M. and Moser, W. O. J. | title=Generators and Relations for Discrete Groups, 4th edition | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9}} 6.5 The binary polyhedral groups, p. 68
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| [[Category:Binary polyhedral groups|Tetrahedral]]
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