group theory

# 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
$A_n$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$D_{n+4}$dihedron,
hosohedron
dihedral group
$D_{n+2}$
binary dihedral group
$2 D_{n+2}$
special orthogonal group
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

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

• the full tetrahedral group is the subgroup of O(3)

$T_d \hookrightarrow O(3)$

which is the stabilizer of the standard embedding of the tetrahedron into Cartesian space $\mathbb{R}^3$;

• the rotational tetrahedral group $T \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to SO(3); this is isomorphic to the alternating group $A_4$;

• finally the binary tetrahedral group is the double cover, hence the lift of $T$ to Spin(3)$\simeq$ SU(2);

• next there is a string 2-group lift $String_{2T} \hookrightarrow String_{SU(2)}$ of the icosahadral group (Epa 10, Epa-Ganter 16)

$\array{ String_{2T} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 T &\hookrightarrow & Spin(3) \simeq SU(2) \\ \downarrow && \downarrow \\ T \simeq A_4 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ T_d \simeq S_5 &\hookrightarrow & O(3) }$

## Properties

### Isomorphisms

The full tetrahedral group is Isomorphic to the symmetric group $S_4$ 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 $A_4$.

### Group order

$\vert T_d\vert = 24$

$\vert T\vert = 12$

$\vert 2T\vert = 24$

### 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
$A_n$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$D_{n+4}$dihedron,
hosohedron
dihedral group
$D_{n+2}$
binary dihedral group
$2 D_{n+2}$
special orthogonal group
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
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.