Binary tetrahedral group

In mathematics, the binary tetrahedral group is an extension of the tetrahedral group "T" of order 12 by a cyclic group of order 2.

It is the binary polyhedral group corresponding to the tetrahedral group, and as such can be defined as the preimage of the tetrahedral group under the 2:1 covering homomorphism:$mathrm\{Sp\}(1)\; o\; mathrm\{SO\}(3).,$where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) It follows that the binary tetrahedral group is discrete subgroup of Sp(1) of order 24.

Elements

Explicitly, the binary tetrahedral group is given as the group of units in the ring of Hurwitz integers. There are 24 such units given by:$\{pm\; 1,pm\; i,pm\; j,pm\; k,\; frac\{1\}\{2\}(pm\; 1\; pm\; i\; pm\; j\; pm\; k)\}$with all possible sign combinations.

All 24 units have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The convex hull of these 24 elements in 4-dimensional space form a convex regular 4-polytope called the 24-cell.

Properties

The binary tetrahedral group, denoted by 2"T", fits into the short exact sequence:$1\; o\{pm\; 1\}\; o\; 2T\; o\; T\; o\; 1.,$This sequence does not split, meaning that 2"T" is "not" a semidirect product of {±1} by "T". In fact, there is no subgroup of 2"T" isomorphic to "T".

The center of 2"T" is the subgroup {±1}, so that the inner automorphism group is isomorphic to "T". The full automorphism group is isomorphic to "S"₄ (the symmetric group on 4 letters).

The binary tetrahedral group can be written as a semidirect product:$2T=Q\; timesmathbb\; Z\_3$where "Q" is the quaternion group consisting of the 8 Lipschitz units and Z₃ is the cyclic group of order 3 generated by ω = −½(1+"i"+"j"+"k"). The group Z₃ acts on the normal subgroup "Q" by conjugation. Conjugation by ω is the automorphism of "Q" that cyclically rotates "i", "j", and "k".

One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) — the group of all 2×2 matrices over the finite field F₃ with unit determinant.

Presentation

The group 2"T" has a presentation given by:$langle\; r,s,t\; mid\; r^2\; =\; s^3\; =\; t^3\; =\; rst\; angle$or equivalently,:$langle\; s,t\; mid\; (st)^2\; =\; s^3\; =\; t^3\; angle.$Generators with these relations are given by:$s\; =\; frac\{1\}\{2\}(1+i+j+k)\; qquad\; t\; =\; frac\{1\}\{2\}(1+i+j-k).$

ubgroups

The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2"T" of index 3. This group and the center {±1} are the only nontrivial normal subgroups.

All other subgroups of 2"T" are cyclic groups generated by the various elements, with orders 3, 4, and 6.

ee also

*binary polyhedral group
*binary cyclic group
*binary dihedral group
*binary octahedral group
*binary icosahedral group

References

*cite book | first = John H. | last = Conway | coauthors = Smith, Derek A. | authorlink = John Horton Conway | title = On Quaternions and Octonions | publisher = AK Peters, Ltd | location = Natick, Massachusetts | year = 2003 | isbn = 1-56881-134-9