Binary octahedral group

In mathematics, the binary octahedral group, name as 2O or <2,3,4> is a certain nonabelian group of order 48. It is an extension of the octahedral group O or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism $\operatorname {Spin} (3)\to \operatorname {SO} (3)$ of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48.

The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism $\operatorname {Spin} (3)\cong \operatorname {Sp} (1)$ where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)

Elements

Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units

\{\pm 1,\pm i,\pm j,\pm k,{\tfrac {1}{2}}(\pm 1\pm i\pm j\pm k)\}

with all 24 quaternions obtained from

{\tfrac {1}{\sqrt {2}}}(\pm 1\pm 1i+0j+0k)

by a permutation of coordinates (all possible sign combinations). All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1).

Properties

The binary octahedral group, denoted by 2O, fits into the short exact sequence

1\to \{\pm 1\}\to 2O\to O\to 1.\,

This sequence does not split, meaning that 2O is not a semidirect product of {±1} by O. In fact, there is no subgroup of 2O isomorphic to O.

The center of 2O is the subgroup {±1}, so that the inner automorphism group is isomorphic to O. The full automorphism group is isomorphic to O × Z₂.

Presentation

The group 2O has a presentation given by

\langle r,s,t\mid r^{2}=s^{3}=t^{4}=rst\rangle

or equivalently,

\langle s,t\mid (st)^{2}=s^{3}=t^{4}\rangle .

Generators with these relations are given by

s={\tfrac {1}{2}}(1+i+j+k)\qquad t={\tfrac {1}{\sqrt {2}}}(1+i).

Subgroups

The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2O of index 6. The quotient group is isomorphic to S₃ (the symmetric group on 3 letters). The binary tetrahedral group, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2O.

The generalized quaternion group of order 16 also forms a subgroup of 2O. This subgroup is self-normalizing so its conjugacy class has 3 members. There are also isomorphic copies of the binary dihedral groups of orders 8 and 12 in 2O. All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).^[1]

Higher dimensions

The binary octahedral group generalizes to higher dimensions: just as the octahedron generalizes to the hyperoctahedron, the octahedral group in SO(3) generalizes to the hyperoctahedral group in SO(n), which has a binary cover under the map $\operatorname {Spin} (n)\to SO(n).$

References

Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups, 4th edition. New York: Springer-Verlag. ISBN 0-387-09212-9.
Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters, Ltd. ISBN 1-56881-134-9.

Notes

↑ Binary octahedral group = $GL_{2}(\mathbb {F} _{3})$ on GroupNames

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