binary tetrahedral group

binary tetrahedral group. In the field of group theory, it is a specific finite group of order 24 that forms the double cover of the rotational symmetry group of a regular tetrahedron. It is isomorphic to the special linear group SL(2,3) and can be realized as a subgroup of the unit quaternions. The group plays a significant role in the classification of finite subgroups of the three-dimensional sphere.

Definition

The group can be concretely defined as the set of 24 unit quaternions given by the eight elements of the quaternion group together with 16 additional elements of the form $(\backslash pm\; 1\; \backslash pm\; i\; \backslash pm\; j\; \backslash pm\; k)/2$. This set is closed under quaternion multiplication, forming a discrete subgroup of the 3-sphere, denoted S³. Equivalently, it is the preimage of the rotational tetrahedral group $A\_4$ under the double covering homomorphism from SU(2) to SO(3). This double cover relationship is central to its structure, with the center consisting of $\backslash \{\backslash pm\; 1\backslash \}$. Another fundamental definition identifies it with the special linear group SL(2,3), the group of 2×2 matrices over the finite field GF(3) with determinant 1.

Properties

With order 24, it is a solvable group but not nilpotent. It is a perfect group, meaning it equals its own commutator subgroup. The group's character table is that of SL(2,3), featuring irreducible representations of dimensions 1, 1, 1, 2, 2, 2, and 3. Its Schur multiplier is trivial. As a subgroup of the quaternions, it provides a tessellation of the 3-sphere into 24 congruent cells, each a regular spherical tetrahedron; this structure is the three-dimensional analogue of a regular polyhedron and is known as the 24-cell. The binary tetrahedral group is also isomorphic to the binary polyhedral group associated with the tetrahedron and is distinct from the binary octahedral group and the binary icosahedral group.

Subgroups and related groups

Its lattice of subgroups is rich. Notable subgroups include the central cyclic group of order 2, the quaternion group of order 8, and several copies of the binary dihedral group of order 8. It contains the normal subgroup known as the dicyclic group of order 12. The full quotient by its center yields the simple group A₄. In the broader classification of finite subgroups of SO(4), it appears as a member of the Dynkin diagram series related to the McKay correspondence. It is a subgroup of the binary octahedral group of order 48 and the binary icosahedral group of order 120. The group is also closely related to the tetrahedral symmetry groups in crystallography and the symmetry of the Hesse configuration in algebraic geometry.

Applications

The group arises in the study of singularity theory, particularly in the classification of Klein singularities of type $E\_6$. In theoretical physics, it appears in the context of string theory and conformal field theory as a discrete symmetry group. Its action on the 3-sphere is fundamental in homotopy theory and the construction of lens spaces. In chemistry, the symmetries described are relevant to the structure of certain complex molecules and quasicrystals. The tessellation of the 3-sphere by its fundamental domain is instrumental in hyperbolic geometry and the study of 4-manifolds through the work of mathematicians like William Thurston.

Category:Finite groups Category:Group theory Category:Symmetry groups