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tetrahedral group

Contents

Idea

The tetrahedral group is the finite symmetry group of a tetrahedron.

As a symmetry group of one of the Platonic solids, the tetrahedral group participates in the ADE pattern:

ADE classification

Dynkin diagram via
McKay correspondence
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A nA_ncyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
D n+4D_{n+4}dihedron,
hosohedron
dihedral group
D n+2D_{n+2}
binary dihedral group
2D n+22 D_{n+2}
special orthogonal group
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

More in detail, there are variants of the tetrahedral group corresponding to the stages of the Whitehead tower of O(3):

String 2T String SU(2) 2T Spin(3)SU(2) TA 4 SO(3) T dS 5 O(3) \array{ String_{2T} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 T &\hookrightarrow & Spin(3) \simeq SU(2) \\ \downarrow && \downarrow \\ T \simeq A_4 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ T_d \simeq S_5 &\hookrightarrow & O(3) }

Properties

Isomorphisms

The full tetrahedral group is Isomorphic to the symmetric group S 4S_4 of permutations of four elements (see Full tetrahedral group is isomorphic to S4).

The subgroup of orientation-preserving symmetries is isomorphic to the alternating group A 4A_4.

Group order

group order:

|T d|=24\vert T_d\vert = 24

|T|=12\vert T\vert = 12

|2T|=24\vert 2T\vert = 24

Group cohomology

The group cohomology of the tetrahedral group is discussed in Groupprops, Kirdar 13.

As part of the ADE pattern

ADE classification

Dynkin diagram via
McKay correspondence
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A nA_ncyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
D n+4D_{n+4}dihedron,
hosohedron
dihedral group
D n+2D_{n+2}
binary dihedral group
2D n+22 D_{n+2}
special orthogonal group
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

References

Discussion in the context of classification of finite rotation groups goes back to:

  • Felix Klein, chapter I.4 of Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, 1884, translated as Lectures on the Icosahedron and the Resolution of Equations of Degree Five by George Morrice 1888, online version

  • Mehmet Kirdar, On The K-Ring of the Classifying Space of the Symmetric Group on Four Letters (arXiv:1309.4238)

  • Narthana Epa, Platonic 2-groups, 2010 (pdf)

  • Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, (arXiv:1605.09192)

See also

Last revised on April 16, 2018 at 02:13:12. See the history of this page for a list of all contributions to it.