Binary tetrahedral group

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_3where "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 angleor 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


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Binary icosahedral group — In mathematics, the binary icosahedral group is an extension of the icosahedral group I of order 60 by a cyclic group of order 2. It can be defined as the preimage of the icosahedral group under the 2:1 covering homomorphism:mathrm{Sp}(1) o… …   Wikipedia

  • Binary octahedral group — In mathematics, the binary octahedral group is an extension of the octahedral group O of order 24 by a cyclic group of order 2. It can be defined as the preimage of the octahedral group under the 2:1 covering homomorphism:mathrm{Sp}(1) o… …   Wikipedia

  • Binary cyclic group — In mathematics, the binary cyclic group of the n gon is the cyclic group of order 2 n , C {2n}, thought of as a extension of the cyclic group C n a cyclic group of order 2.It is the binary polyhedral group corresponding to the cyclic group.ee… …   Wikipedia

  • Tetrahedral symmetry — A regular tetrahedron has 12 rotational (or orientation preserving) symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation.The group of symmetries that includes reflections is isomorphic to S 4 …   Wikipedia

  • Group action — This article is about the mathematical concept. For the sociology term, see group action (sociology). Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle acts on the set of vertices of the… …   Wikipedia

  • Dicyclic group — In group theory, a dicyclic group (notation Dicn) is a member of a class of non abelian groups of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di cyclic. In the… …   Wikipedia

  • Quaternion group — In group theory, the quaternion group is a non abelian group of order 8. It is often denoted by Q and written in multiplicative form, with the following 8 elements : Q = {1, −1, i , − i , j , − j , k , − k }Here 1 is the identity element, (−1)2 …   Wikipedia

  • Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia

  • nitrogen group element — ▪ chemical elements Introduction  any of the chemical elements that constitute Group Va of the periodic table (see Figure >). The group consists of nitrogen (N), phosphorus (P), arsenic (As), antimony (Sb), and bismuth (Bi). The elements share… …   Universalium

  • Point groups in three dimensions — In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”