Tetrahedral Symmetry - Chiral Tetrahedral Symmetry

Chiral Tetrahedral Symmetry

T, 332, +, or 23, of order 12 - chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D₂ or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A₄, the alternating group on 4 elements; in fact it is the group of even permutations of the four 3-fold axes: e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23).

The conjugacy classes of T are:

• identity
• 4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321)
• 4 × rotation by 120° anti-clockwise (ditto)
• 3 × rotation by 180°

The rotations by 180°, together with the identity, form a normal subgroup of type Dih₂, with quotient group of type Z₃. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.

A₄ is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d: the group G = A₄ has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C₆ or D₃, but neither applies.