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:$mathrm\{Sp\}(1)\; o\; mathrm\{SO\}(3).,$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 octahedral group is discrete subgroup of Sp(1) of order 48.

Elements

Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units:$\{pm\; 1,pm\; i,pm\; j,pm\; k,\; frac\{1\}\{2\}(pm\; 1\; pm\; i\; pm\; j\; pm\; k)\}$with all 24 quaternions obtained from:$frac\{1\}\{sqrt\; 2\}(pm\; 1\; pm\; 1i\; +\; 0j\; +\; 0k)$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 2"O", fits into the short exact sequence:$1\; o\{pm\; 1\}\; o\; 2O\; o\; O\; o\; 1.,$This sequence does not split, meaning that 2"O" is "not" a semidirect product of {±1} by "O". In fact, there is no subgroup of 2"O" isomorphic to "O".

The center of 2"O" 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 2"O" has a presentation given by:$langle\; r,s,t\; mid\; r^2\; =\; s^3\; =\; t^4\; =\; rst\; angle$or equivalently,:$langle\; s,t\; mid\; (st)^2\; =\; s^3\; =\; t^4\; angle.$Generators with these relations are given by:$s\; =\; frac\{1\}\{2\}(1+i+j+k)\; qquad\; t\; =\; frac\{1\}\{sqrt\; 2\}(1+i).$

ubgroups

The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2"O" of index 6. The quotient group is isomorphic to "S"₃ (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 2"O".

The generalized quaternion group of order 16 also forms a subgroup of 2"O". 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 2"O". All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).

ee also

*binary polyhedral group
*binary cyclic group
*binary dihedral group
*binary tetrahedral 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