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:

Dynkin diagramPlatonic solidfinite subgroup of $SO(3)$finite subgroup of $SU(2)$simple Lie group
$A_l$cyclic groupcyclic groupspecial unitary group
$D_l$dihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
$E_6$tetrahedrontetrahedral groupbinary tetrahedral groupE6
$E_7$cube/octahedronoctahedral groupbinary octahedral groupE7
$E_8$dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8

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

Dynkin diagramPlatonic solidfinite subgroup of $SO(3)$finite subgroup of $SU(2)$simple Lie group
$A_l$cyclic groupcyclic groupspecial unitary group
$D_l$dihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
$E_6$tetrahedrontetrahedral groupbinary tetrahedral groupE6
$E_7$cube/octahedronoctahedral groupbinary octahedral groupE7
$E_8$dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8